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LOGARITHMIC* ARITHMETIC* 

CONTAINING A NEW .AU^OQRRECT 

TABLE OF Lof^THMS 

OF THE NATURAL NUMBERS FROM 1 TO 10.000, 

EXTENDED TO SEVEN PLACES BESIDES THE INDEX J AND SO 
CONTRIVED, THAT THE LOGARITHM MAY BE EASILY 
FOUND TO ANY NUMBER BETWEEN 1 AND 
10,000,000. 

ALSO, 

AN EASY METHOD 

OF CONSTRUCTING A TABLE OF LOGARITHMS, TOGETHER 

WITH THEIR NUMEROUS AND IMPORTANT USES 

IN THE MORE DIFFICULT PARTS OF 

ARITHMETICK. 

TO WHICH ARE ADDED A NUMBER OF 

ASTRONOMICAL TABLES, 

BY WHICH THE DIFFERENT PHASES OF THE MOON,— THE 

TIMES OF HER OPPOSITION AND CONJUNCTION, MAY 

BE COMPUTED WITH THE OREATEST EASE AND 

exactness: and an easy method 

of calculating £. 

SOLAR AND LUNAR ECLIPSES j * 

V " 

ILLUSTRATED WITH 

.*•■•.. 

GEOMETRICAL PROJECTIONS: 

DESIGNED FOR THE INSTRUCTION OF YOUTH IN THE 

SCHOOLS AND ACADEMIES 

OF NEW ENGLAND, 

BY ELIJAH HINSDALE JURRITT. 
WILL I A MSB VR Iff} ; 

PRINTED BY KPKRAIM WHITMAN. 
161& 









"7*) 



DISTRICT OF MASSACHUSETTS, to wit * 

District Clerk* s*Qj/itt. 
Be it remembered, That on the fifteenth day of 
September, A. D. 1818, and in the forty-third year of the Inde* 
pendence of the United States of America, Elijah Hinsdale 
Burritt, of the said district, has deposited in this office the 
title of a book, the right whereof he claims as author, in the 
words following, to wit ; 

Logarithmick Arithmetick, containing a new and correct Ta* 
ble of Logarithms of the natural numbers from 1 to 10,000, ex* 
tended to seven places besides the index ; and so contrived, that 
the Logarithm may be easily found to any number between I 
and 10,000,000. Also, an easy method of constructing a Table 
of Logarithms, together with their numerous and important 
uses in the more difficult parts of Arithmetick. To which are 
added a number of Astronomic! Tables, by which' the differ- 
ent phases of the Moon, the times of her opposition and con- 
junction, may be computed with the greatest ease and exact- 
ness : and an easy method of calculating' Sqlar aud Lunar 
Eclipses ; illustrated with Geometrical Projections : Designed 
for the instruction of Youth in the Schools and Academes of 
New England. By Elijah Hinsdale Burrjtt. 

In conformity to the act of the congress of the United States, 
entitled, " An act for the encouragemeut of learning, by secur- 
ing the copies of mapsj charts and books, to the authors and 
proprietors of such copies, during the times therein mentioned s" 
and ahso to an act entitled, « An act supplementary tp ajiact, en- 
titled, an act for the encouragement of learning, by securing 
the copies of maps, charts and hooks, to the authors and propri- 
etors of such copies during the timrs therein mentioned ; and 
^xtentling the benefits thereof to the arts of designing, engraving 
and etching historical, and other prints." JOHN W. DAVIS, 
Clerk qf the District qf Massachusetts, 



J 



, 1 



dak 



FREFA 



« TlltStask cfau author ;* says Dr. Johnson, is to teach whas» 
Is not known, or to recommend known truths by his manner of 
adorning them." To attempt the former is sufficiently irksome 
to enervate endeavour, and to succeed in the latter, he nvust sur- 
mount obstcles that no sagacity can avoid, and. encounter diffi- 
culties which prevision cannot alleviate. He must appeal t* 
judges prepossessed by passions or biased by prejudices : to 
some who tire more willing to go wrong by their own judgment, 
than to be indebted lor a better or a safer way to the wisdom of 
another; and toothers, whose good taste and perspicacity will 
not allow them to read any thing until its reputation is ectablish- 
ed by the " Ministers of Criticism." But of those who are 
jnor*c inclined to be pleased* who may be persuaded to remit 
their censure at those errdurs into which tho. author has inad- 
vertently or unavoidably fallen, and who will pass with a friend- 
ly eye over those imperfections that are inseperably connected 
"with all human productions ; it is hoped* though the reader 
should not find his feelings ifhperceptably stolen from him by 
the enchantment of novelty, or his understanding captivated by 
the profundity of invention, that if his patience can endure a care- 
ful perusal of the following work, be will dismiss it with, at least 
this «' negative encomium," that it is a laudable attempt to im- 
prove the instruction of the unlearned, and furnish schools in 
genera! with a useful and comprehensive system of Practical 
Arithmetick. 

He who is resolved to attain any degree of eminence in thje 
science of Mathematicks, dooms himself to wade through a toil* 

* some* course of severe, uriinte re sling labour, unaccompanied 
with any of those charms that can engage the feelings by their 
novelty, or delight the imagination by their lusture ; without 
any hope of honorable reward to stimulate to exertion, or sof- 
ten the asperities of study. But the writer of Common Arith- 
metick has a much easier taak* The labours of many that have 

" gone before him as " pioneers,*' who wore distinguised by pro- 
found investigation and mathematical researches, have render- 
ed his path comparatively easy, even where their feet have never 
trodden ; " as the sun dissipated the shades of darkness, and 
spreads a twilight beyond the immediate influence of his direct 
beams." Indeed, it may be said that the only necessary rcsourse 
of the writers of the present age> is to copy the best examples 
of their "predecessors with such modifications and amendments 



I 



PREFACE. 



as the different modes of reckoning in business, the fluctuation 
of coins, weights, and measures, and as the invention of means 
to facilitate the samfltfypgpns require. 

The practical sysf^Hpr Vulgar Arithmetick already extant 
are very numerous ; ana the authors of some of them have ex- 
hibited great knowledge and ability, it may therefore be ex- 
pected, that he who ventures to add to their number should pro- 
duce a substantial plea for such an obtrusion on the publick no- 
tice. It may justly be expected that he should be either an ac- 
curate schollar or a profound mathematician. The writer has 
no claim either to the one or the other of these distinctions ; 
' and must therefore plead another apology. 

Having been repeatedly solicited by a number of gentlemen 
respectable for their understanding and skill in Mathematical 
Science, to publish his Table of Logarithms constructed only 
for private use, together with their numereus and important 
uses in the more difficult parts of Arithmetick Trigonometry 
&c. and believing', notwithstanding the endless variety of books 
now extant on Vulvar Arithmetick, there is yet room to intro- 
duce usefulness with novelty, the author of the following work 
humbly ventures to*assume the responsibility of endeavouring to 
avoid the redundencies of some, and to supply the deficiencies 
of others. 

It is believed that a more accurate and extensive System of 
Logarithms for natural numbers was never before published in 
this country* The best American Tables are carried only to 
six places besides the index, which, though capable of giving an- 
swers sufficiently exact in most solutions, are, notwithstanding, 
deficient where great accuracy is required. Iff constructing 
the following table the author has pursued the" Differential 
Method** hinted at by Mr Henry Briggs, Professor of Geometry 
in Gresham College London, and Dr. Hutton's " Practical Rule 
for the Construction of Logarithms." 

He was also solicited to publich with this, a Table of Loga- 
rithmick Signs, Tangents, &c. carried to seven places decimal, 
which should correspond with those for natural numbers; but 
as thb would render the work too voluminous for a common 
school-book, it was thought unadvisable to augment the expense 
-without proportionably increaseing the advantage. Should their 
occur sufficient reason for publishing such a system of Logar- 
ithms and Logarithmick Sines and Tangents, together with oth- 
er Logarithmick and Mathematical Tables, they may be given 
to the publick at some future period. 

The method op commuting jby Logarithms, whepe it can 
be adopted, aa in the evolving of roots, is manifestly the most 
expiditious of any that human powers have hitherto invented. 
*TV learner, who before was unacquainted with their propevr*- 



PREFACE. 

ties, Who has had the patience and assiduity to toil through the 
tedious course of evolution by the usual process, will admire to 
find that so much labour should ever ^Mpreduced to so much 
ease ! ■ - .^F 

The author has endeavoured, though in some instances at the 
expense of deviating from Ancient usage, to arrange the sever- 
al parts of Vulgar Arithmetick according to their relative im- 
portance, and their mutual dependence upon each other ; and 
to render the whole as easy and familiiar as the nature of the 
subjects would admit. 

From what experience the author has had in the instruction 
of youth, aud the general complaint of the want of some work 
adapted to accompany those elementary treatises on Astronomy 
which are used in our Schools and Academies, the writer was in- 
duced to subjoin a short* though imperfect sketch of Practical 
Astronomy; And in this, he has been more desirous to be use- 
ful than to appear oiiginal. AH the Tables except the II, XVIII 
and XIX, together with the method of calculating the time of 
New and Full Moons and Eclipses, were taken irom Dr. Brew- 
ster's edition of Fugeson's Astronomy lately published. The 
method of Projecting Eclipses is purely mathematics, and is 
not subject to those inacuracies which the use of the sector is 
liable to produce. 

Whatever merit justice may award to the following primetial 
pages, the author cannot forbear repeating, that it was the hope 
only of being useful, that induced their submission to publick 
scrutiny. If he has failed in the attempt, the disquietude of 
disappointment will -be allayed by the reflection, that he is not 
the first " gut magma ecocide t ausia" who have misjudged in 
their abilities. To suppoae the work wholly exempt from *r- 
rours> whether of the press or of -the pen, would be to suppose 
what the most vigilant care has seldom performed. And wheth- 
er these will be found to have a counterbalance of good, is re- 
fered to the candour of those who may read to decide ; — they 
are the constituted judges, and to them he submits with defter- 
ence and respect. 

R H. B. 
Williams College, October, 181*. 



NOTATION ..--.* 

Simple Addition - * - -• 

, .Federal Money # - . * 

Simple Subtraction - - - - 

Subtraction of Federal Money- • «. .# 

Simple Multiplication * - . „ * , * 

Multiplication of Federal Money - - 

Simple Division - * * -. 

Contractions in Division - - - — 

Short Division - - - - - 

.Supplement to Multiplication - - - 

Practical Questions in Multiplication and division 
Compound Addition - . - •• * 

Compound Subtraction - - - - 

Compound Multiplication - - •■ • - 

-i Of Weight) MeasuiCj Sterling Money Sec. 

Compound Division - - - ' * 

— » — Of Sterling Money, Weights, Measures, &c. 

Redaction - . - - 

Reduction of Currencies 

Table of Coins current in the United States 

Duodecimals* or Cross Multiplication 

Simple Proportion or Rule of Three * 

Of Logarithms -- - 

Construction of Logarithms * 

Another Method of Computing Logarithms 

Directions for taking Logarithms arid their 

numbers from the Table ' - 
Method of Calculating by Logarithms - * 

Multiplication by Logarithms * * - 

Division by Logarithm* -' - - •■ - 

Proportion by Logarithm* > • - - - ' - • * 
Arithmetical Complement * - * 

Vulgar Fractions - «• -> - «* 

Decimal Fractions „---•* 
Addition of Decimals 
Subtraction of Decimals - 

Multiplication of Decimals - - 

' Diyision of Decimals - 

'Reduction of Decimals - 

Simple Interest - - - 

Insureance, Commission, and Brokerage 
Discount - - - - - T - - 

$quation of Payments ------ 

IPeS'lowship - 

Compound Fellowship - - - 

Alligation - - 

;Tare and Trett - - 

Compound Proportion) or Doupje Rule of Three 

Do. by Logarithms - - - 



Pag&+ 
9 

* lO 
li 
IS 
13 
14 
15 
16 
17 

- YS 

- 20 
SO 

- 21 
24 
26 

27 

26 
39 
30 
33 
34 
35- 
38 
44 
47 
49 

50- 

5? 

52 

53 

54 

55 

36 

58 

59 

60 

61 • 

62 

63 

65 

69 

99 

70 

71 

72 

73 

71 

77 

79 



CONTENTS $ 

Compound Interest - ~ - 80 

Involution - - - - - - - 82 

Evolution, or Extraction of Roots ' -,,' ' -' - - 83 

Do. by Logarithms ------ 8f. 

Practical Questions in Evolution and Involution - 94 

Annuities, or Pensions - 100 

Vulgar Fractions - - - IQ5 

Reduction of Vulgar Fractions, - - - .- 106 

Addition of Vulgar Fractions - . - - MO 

Multiplication of Vulgar Fractions - - - lit 

Division of Vulgar Fractions - - - - -Hi 
Simple Proportion in Vulgar Fractions - - 112 

Reduction of Decimal Fractions - - - - 1M 

Simple Proportion by Decimals . - . - -. - 115 

Simple Interest by Decimals - - ■» - 1 1 & 

Tables shewing the Amount, and the Rebate of one Dollar, 

at 6 per cent for Years and MouUra - - * 119 
Construction of said Tables • -* - -■ - 120 

Compound Interest by decimals - - - 122 

Logarithmick Tables for Years - - - - 123 

Do'. do. for Months and Da> s - - 124 

Arithmetical Progression - - - 125 

Creometrica I Progression - ... - 12T 

Position - - ■ - - - - -• - 133 

Double Position 134 

Permutation of Quantities - - - 1 23 

Construction of Tables belonging to Compound Interest 130 
Tables relating to Compound Interest . - - 139. 

Practical Astronomy - - - - « -. 4 £3 

Tables for Calculating the True Time of New and Full 

Moon's an4 Eclipses * 149—165 

Precepts relating to the preceding Tables - - 165 

Do. for Calculating the Uptime of New or Full Moon 165 
To Calculate the true place of the Sun for any given Moment 

of Time - - - - - 186 

To know whether there is an Eclipse at the time of any New 

or Full Moon - - - - -.18* 

To Project an Eclipse of the Sun - <■ * 19^ 

To Project an Eclipse of the Sun Geometrically - \9% 

Geometrical Projection of Eclipses - - - - 19$ 
To preject an Eclipse of the Moon - - . 196 

To project an Eclipse of the Mooyi Geqmetrically * 200 
To find the number of Eclipses, there are in any given Year, 

and in what Months they happen - - SQ3 

To find on what part of the Globe any given Eclipse of the 

Sun or Moon is visible - ' - - - 204 

Table of Logarithms for Natural numbers - 207 — 24f 

Appendix to Logarithms - - * - *XVi— ^A 



Explanation of Characters used in this book. 

«= Equal to, as J 2d. =* Is. signifies that 12 pence are equal to 

1 shilling. 

+ More, the sign of addition, as 5 -f 7=12, signifies that 5 
and 7 added together, are equal to 12. 

— - Minus, or less, the sign of subtraction, as 6 • — 2=4, signi- 
fies that 2 subtracted from 6, leaves 4. 

X Multiply, or with the signof Multiplication ; as 4 X 3=12 
signifies that 4 multiplied by 3, is equal to 12. 

-r- The sign of division ; as 8 -5- 2 « 4, signifies that 8 divided 
by 2, is equal to 4 ; or thus, i=4, each of which signifies 
the same thing. - 

:: Four points set in the middle of four numbers, denote them 
to be proportional to one another, by the rule of three ; as 

2 : 4 :: 8 : Id ; that is, as 2 is to 4, so is 8 to 16. 

4/ Prefixed to any number, supposes that the squre root of that 
number is required. 

3 

tf Prefixed to any number, supposes the cube root of that num- 
ber is required. 

4 . 

4/ Denotes the biquadrate root, or fourth power, Sec. 

MULTIPLICATION TABLE. 



1 | 2 | 3 | 4 


5 | 6 | 7 | 8 | 


•1 


10 | 


" 1 


12 


2 | 4 | 6 | 8 1 10 j!2 [14 |l6 | 


18 1 


20 


22 


24 


3 | 6 • 9 |12 |i5 |*8 |2i |24 


a7 | 


30 


33 


36 


4 | 8 |I2 |i6 1-0 |24 |28 |32 


36 


40 


44 


48 


5 jlO 15 (20 


25 |30 . 


36 (40 


, 4S 


.50 


1 » 


60 


6 |»2 


18 


^4 


3u |36 |42 |48 


54 


60 


| 66 


72 


7 


14 

it 


21 
24 


28 
32 


35 

lo" 


42 149 |56 1 
48~|56~l4i4"| 


63 I 


70 


77 


I 84 


8 


72 | 


80 


88 


96 


9 


18 


27 

So" 


36 

40 


45 
50 


54 

60 


63 1/2 


8. 


90 


9V 


108 


10 1 


20 


70 |80 


90 


100 


L.10 


120 


1 1 


22 


33 
3(T 


44 

Is" 


55 

60~ 


66 
72 


77 

84" 


88 


99 


UO 


121 


132 


|,2 


24 


J6 


108 


120 


132 


144 



'To learn this table : Find your multiplier iri the left hand col- 
umn* and the multiplicand a-top, and in the common angle of 
meeting, or against your multiplier, along at the right band, and 
under your multiplicand, you will find the product, or answer.. 



WABLE£ OF WEIGHTS AND MEASURES, 

1. Sterling Money. 

4 farthing* make 1 pevny, tf. 

11 pence, 1 shilling, *. 
20 shillings, 1 pound, £ 

2. Troy Weight. 

34 grains (gr.) make 1 penny-weight, marked fivft. 

20 penny-weights, 1 ounce, zu. 

12 ounces, 1 pound,, id. 

3. Avoir dufiois Weight. 

16 drams (dr.) make 1 ounce, wz. 

16 ounces, 1 pound, lb, 

28 pounds, I quarter of a hundred weight, ?r. 

4 quarters, 1 hundred weight, owf. 

20 hundred Weight, 1 ton, T. 

By this weight are weighed all coarse and drossy goods, grc 
eery wares, and all metals except gold and silver. 

4. Cloth Measure. 

4 nails fna.J make 1 quarter of a yard, qr. 

4 quarters, 1 yard, yd. 

3 quartters, • 1 EH Flemish, E. >V. 
s quarevs, 1 Efl English, E. SB. 
6 quarters, 1 Ell French, A Fr. 

5. Dry Measure. 

2 pints (fit.) make 1 quart, qt. 

5 quarts, 1 peck, fik. 

4 pecks, l bushel, bu. 

This measure is applied to grain, beans, flak-seed, salt, roof 
oysters, coal, &c, 

6. Wine Measure. 

4 gills (gL) make 1 pint, fit. 

3 pints, 1 quart, qt. 

4 quarts, 1 gallon, gal, 
311 gallons, 1 barrel, bl. 

42 gallons, 1 tierce, Her. 

43 gallons, 1 hogshead hhd. 

2 hogsheads, 1 pipe, fi. 

3 pipes, l tun, T. 
Al* brandies, spirits, mead, vinegar, oil, Sec. are measured fr 

wine measure. Note. — 231 solid inches, make a gallon. 
7. Long Measure. 

3 barley corns (b. c.) make 1 inch, marked fa, 

12 inches, 1 foot, ft. 

3 feet, 1 yard, yc \. 

Si yards, 1 rod, pole, or perch, rd. 

40 rods, 1 furlong, fur. 

8 furlongs, 1 mile, m. 

3 miles, 1 leagye, i ca . 
49$ statute miles, 1 degree on the earth. 
340 degrees, the circumference of the cm'A* 



8 TABLES OF WEIGHTS AND MEASURE*. 

The i|se of kw^ measure is to measure the distance of places, 
or any other thing, w^here length is considered, without regard 
to breadth. 

N. B. In measuring the height of horses, 4 inches make t 
hand. In measuring depths, 6 feet make 1 fathom, or French 
toise. Distances are measured by a chata, four rods long, con- 
taining one hundred links. 

8. Land or Square Measure. 
144 square inches make 1 square foot. 

9 square feet, 1 square yard, 

30$ square yards, or ) , 

272^ square feet, J * squ * r9 10fl ' 

"40 square rods> 1 square rood. 

i 4 square roods, 1 square acre 

| 640 square acres, I square mile. 

i 9 S9iid or Cubic Measures 

\ 1728 solid inches make 1 solid foot. 

f « r ee ! °l u 0Und • ' im u "' ° r I » ton or load. 

50 feet of hewn timber, y 

\ 128 solid feet, or 8 feet long, > • v cord of wood; 

•«. 4 wide, and 4 high, 3 

1 All solids, or things that have length, breadth and depth, are 

I measured by this measure. N. B- The wine gallon contains 

J 231 solid or cubic inches, and the beer gallon, 232, A1)Ushet 

i contains 2150,42 solid inches. \> 

f 10. Time. 

60 seconds (S. ) make 1 minute, marked 8. Mi. 

*0 minutes, 1 hour, h > 

54 hours, I day, <*• 

7 days, 1 week, w. 

4 weeks, 1 month, mo. 

}3 months, 1 day, and 6 hours, 1 Julian year yr. 

Thirty days hath September, April, June, and November, 
I February twenty-eight alone, all the rest have thirty-one. 
{ N. B. In bissextile or leap-year, Febmary hath 19 days. 
J 1 1 Circular motion. 

\ 60 second's (") make 1 minute, marked r 

: «0 minutes, 1 degree, # 

! 30 degrees 1 sign, *• 

{ '\% sign?, or 560 degrees, the wh*lc great circle of the Zodiac. 



ARITHMETICS 



»o«< 



i 



S THE Science of Numbers, and exhibits the method, or 
art of computing by them : it is divided into five parts, viz. 
Rotation, Addition, Subtraction, Multiplication, and Division. 

NOTATION. 

Notation teaches how to read and write numbers represent* 
ed by the following characters, 1, 2, 3, 4, 5, 6, 7, 8, 9, O ; each 
of which has a simple value, and also a local value, according to 
the order pf their combination, as in the following table. 




I 



10 VULGAR 

To know the value of any number tfjiguret. 
RULE. 
Numerate from the right hand figure to the left as in the ta- 
ble, and to the simple value of each figure apply its local value, 
according as it is removed from the place of units towards the 
right. * 

EXAMPLES. 

Mead the folio wing numbers. 
75 Seventy five. 
937 Nine hundred and thirty seven. 
46 1 7 Four thousand six hundred and seventeen. 
59000028 Fifty nine millions and twenty eight. 
679244321 Six hundred and seventy nine millions two hun- 
dred and forty four thousand three hundred and twenty one. 
To write numbers. 
. m RULE. 

Beginning on the right hand, write unites in their place, 
tens in the tens place, and so on towards the left hand, writing 
each figure according to its value in numeration, and supplying 
those places of the natural order with cyphers which are omitted 
in the question. 

, EXAMPLES. 

Write down the following numbers. 

Twenty four 24 

V Three hundred and five 505 

Seven thousand and ninety 7090 

Eight millions and eight 8000008 



SIMPLE ADDITION. 

Is the uniting into one number several smaller ones tf the 
same denomination. 

RULE. 

Write the numbers distinctly, units under units, tens under 
tens, &c. Then reckon the amount of the right hand column. 
If it be less than ten, write it down. If it exceed ten, write the 
units only and carry the tens to the next place. 

In like manner carry the tens of each column to the next, 
observing to write down the full sum of the left hand column. 
METHOD OF PROOF. 

1. Draw a line between the first and second line* of figures 
to cut qfF the first number. 

2. Add the other numbers together and set their sum Under 
the sum of all the number*. 



ARITHMETIC*. n 

3% Add the numbers last found with the numbers cut off, and 
if their sum be the same as that of the first addition, the work 
is right. 

EXAMPLES. 

CO «• < 0> 

4 9643 . S 2 1 4 4 3214 



645 3 2 13100 1960 
98760 19264 420 
14000 83400 40 



Sum 226935 14 7 778 45634 Sum 



17. 7292115764 2420 



Proof 2 2 69 3.5 147 778 4563 4 Proof 



4. Add 86S5, 2194, 7421, 5063, 2194, and 1245. An*. 26754. 

5. How many days in the 12 calender months? An*. 365. 

6. From the building of Rome to the death of Alexander, was 
683 years, from the death of Alexander to the Christian era, was 
65 years ; how many years from the building of Rome to the 
present date, 1818? Ann. 2566. 

At the late census of 1810, the inhabitants of New England 
were as follows. 

Of District of Maine 228,705, New Hampshire 214,460, 
Vermont 217,896, Massachusetts 472,040, Rhode-Island 76,931, 
and of Connecticut 261,942^ what was the sum total ? 

Ana. 1,471,974. 



FEDERAL MONEY. 

By an act of Congress, all the accounts of the United States, 
the Sallaries of Officers, all Revenues, fee. are to be reckoned 
in Federal Money, which mode of reckoning in point of simpli- 
city is the nearest allied to vfhole numbers, as it increases like 
<them in ten fold proportion. 

ADDITION OF FEDERAL MONEY. 

RULE. / 

Write the numbers according to their value, that is, dollars 
under dollars, dimes under dimes, cents under cents, mills 
under mills, &c. then proceed the same as in whole numbers, 
observing to place the seperatrix in the sum total directly under 
the seperating ptints above. 



i 



I* 









VULGAR 














EXAMPLES. 


._ ♦ , 








< l l 




. w , 




f«) 




s 


D. C. 


M. 


8 D. C. M. 


8 


D. C. 


M. 


365 


5 4 


l 


439 3 4 


136 


5 1 


4 


487 


6 





416 3 9 


125 


9 





694 


6 7 





168 9 3 4 


$00 


9 


» 


439 


6 


fr 


239 6 


304 





6 


742 


5 d 





143 5 


111 


1 9 


1 



Ana. 272* 8 6 1406 6 9 3 877 7 1 .*/«. 



4. What is the sum total of 8 $67,14, g 117,09, 837,75, 
S 11,05, and 8 96,50? Ana. 8 629,53. 

* 5. What is the amount of 8 10$,12* cts. 8 9,67, 45 cts. 67 
cts. and 8 1 ,08 cts. f Ana. 8 1 13,99,5. 

6. Received of A. B. and G. a sum of money*; A. paid 
8 123,08 cts* B. paid four times as much as A. and C. paid as 
much as A. together with twice as much B. Required the sum 
of payments? Ana. 8 1723,12 cts. 



SIMPLE SUBTRACTION. 

Is the operation by which we take a lesser number frtm a 
greater of the same denomination, and thereby find their differ- 
ence, or remainder. 

The lesser number is called the subtrahend, the greater 
number the minuend, and the number found by the opera- 
tion the REMAINDER, Or DIFFERENCE. 

rule: 

1. Place the less number under the greater, so that units may 
stand under units, tens under tens, &c. and draw a line under 
them. 

2. Beginning at the right, take e^ch figure in the subtrahend 
from the figure over it, and set the "remainder under the line. 

3. If the lower figure be greater than the ©ne over it, add ten 
to the upper figure, from which figure so increased, take the 
lower, and write the remainder, carrying one to the next figure 
in the lower line, and thus proceed till the whole is finished. * 

PROOF. 
Add the remainder to the less number, and if the sum be 
equal to the greater, the work is right. 

* If the cents are less than 10 place a cypher in the tens place, 
or place of dimes; example, write 10 dots, and 3 cents, 8 10,03, 



ARITHMET1CK. 



*9 



EXAMPLES. 



From 
Take 


H8r 6 2$ 
2 3 4 3 7 5 6 


; £rom 
Take 

Rent. 

Ptoof 


. r (2) 

5 3 2 7 4 6 7 

10 8 4 3 8 


Item. 


9 4 3 8 6 9 


4 3 19 2 9 


Proof 


3 2 8 7 6 2 5 


5 32 7f4i <6 7 



S. From 2637804, take 2376982. sAn*. 26082?; »' 

4. From 3762162, take 826541. ^n«. 2935621. 

5. From 78213666, take 27821890. An*. 50391716. 

~ 6. From ninety seven thousand six hundred and thirteen, 
subtract thirty thousand nine hundred and nine* An*. 66704. 

7. From the destruction of Carthage to the year of our Lord 
1818, was 1965 years, how long before the Christian era was 
the city, destroyed I An*. 147 years. 

8. Gun powder was invented by a Monk of Cologne,, in 1330, 
how long has4t been in use to this date 1818 ? 

An*. 4&8 yean. 

9. The Arabian method of Notation was first known in Eng- 
land in the year 1150 ; how long thence to 1818 ? 

An*. 668 years. 
SUBTRACTION OF FEDERAL MONEY, 

* RULB. 

Place the numbers according to their value, aiid subtract as 
in whole numbers. 



EXAMPLES. 



0) 

8 D. C. M. 

From 48 9 6 4 
Take 37 8 9 S 



(*>-"V 
8 D, C. M. 

From 940 6* 4 9 

Take - 804 9 6 3 



An*. 11 7 1 



An*. - 135- 6 8 6 



From 125 dols: take 9 dols. and 9 cts*. An*, g 115 91 cts. 

From 127 dols. 1 cent, take 8 41 10 cts. An*. 8 85 91. 

From 365 dolls. 12 cts. take $ 18T 3$ cts. An» t .t'\ff 77. 

From 100 dolls, take 99 cts. An*, g 9£ 01 cent. 

A. owes B. 1000 dols. and pays him in part, as follows* 

at onetime 8 236 10 cts. at another 8 108 25 cts. and lets 
him have fifty bushels of wheat at 2 dollars pr. bushel, together 
with a horse worth 85 dols. and a plated harness valued at 8 <•>$ 
75 cts. 5 it is required to find how much is yet due ? 

•" An*. S 409 90 cts. 



o. 
4. 
5. 
6. 
7. 
viz.* 



14 



VULGAR 



SIMPLE MULTIPLICATION. 



Is the operation by which we increase, or repeat One of two 
numbers of the same denomination, as often as there are unks 
in the other. 

The number to be multiplied, is called the multiplican». 
The number by which we multiply, is called the multipier. 
The number found by the operation, is called the propuct. 
The Multiplicand and Multiplier, arc both called factors. 

RULE. 

I Place the Multiplier under the Multiplicand), so that uniu 
may stand under units, tens under tens, &c. and draw a line un- 
der them. 

2. Begin at the right, multiply each figure in the Multiple 
oand by the Multiplier, carry one for every ten, and you w|Jl 
have the product. 

PROOF. 

Multiply the Multiplier by the Multiplicand. 

EXAMPLES. 

0> .;..., . (2) 

Multiply 436 Multiplicand. Multiply 90036 

by 5 Multiplier. .by: 7 



2180 Product. 



(3) 
Multiply 34293 
by 74 

137172 
- 240051 



. . Am v 630252 

Multiply 32745654473 
by' 234 



Ans. .2537682 



1309826.17892 
98236963419 
6,5491 $08946 

A . ". ' „-.j"^ i ji i ■ i 
Product 7662483 1466&2 



L 



5. Multiply 364111 by 56. . Ana. 20390216. 

6. Multiply 7128368 by 96. 4m. 684323328. 

7. Multiply 123456789 by 1440. Ane. 177777776160, 

CONTRACTIONS. 
When there are cyfihers on the right of one, or both the 
factors: 

RULE. 
Proceed as before. Multiply by the significant figures, neg- 
lecting the cyphers, and on the right of the product place as 
many cyphers as were neglected in both the factors. 



ARITHMETICS.. 15 

s 
EXAMPLES. • .'.' 

1. Multiply 1234*00 (2) 

by 7500 360000 by 120000a 
" ■ ■ • , 1200000 



61725 '• : 

86415 432300000000 An: 



Product. 9258750000 



3. Multiply 461200 by 72000. Ana. 33206400*0*,. 

Multiply 815036000 by 70300 Am. 57297030800000. 

MULTIPLICATION OF FEDERAL MONEY. 

RULE. 

Multiply the given price by the quantity) and carry as in 
whole numbers. The seperatrix will be as many figures from 
the right hand in the product, as in the given price. 

examples. 
). What wHl 36 yds. of broad cloth come to at six dollars 
twenty fire cents pr. yd. ? 

8 6 25 

36 yards. 



37 50 
187 5 



Am. % 225 00 225 dolls. cents. 



2. What cost 15 yds. calico at 67 cts. pr. yd. ? An*, g 10 0$. 

3. Wbat cost SS yds. kerseymere at g 1 67 cts. pr. yd. ? 

Ana. g 58 45. 

4. What is the value of 75 yds. satin at 2 3 75 cts. pr. yd! ? 

Ans. 8281 25. 

5. What tost 131 bushels wheat at 8 I 67 cts. per bush. > 

Ana. 8 218 77. 
6 What cost 126 gallons molasses 32 cts. per gal. ? 

An 8. g 40 96. 
7. What will 66 bushels oats cost at 28 cts. pr. bush. 

Ana. g!8 48. 
& What cost 97 lb. sugar at 1 2* pr. lb. ? Am, g 12 12$ 



1 



VURGAR 

Mr. William Williams, 

Bought of James ftfcrcb&if. 
10 ibs. Green Tea, at g 2 15 cts. pr. lb. 

36 » Coffee, 



15 " Loaf Sugar, 
o cwt. of Malaga Raisins, 
36 gallons Wine, 
91 " Molasses, 



21*cts.pr. lb. 

25 cts. pr. lb. 
7 00 pr.cwt. 
2 12 cts. pf. gal. 

36 cts. pr. gal. 



Received payment. 
New-York, the 1 st May, 1 8 18. 



g 136, 59 



James Merchant. 



SIMPLE DIVISION. . 

Teaches to find how often one number is contained in another 
of the same denomination; or to find a quotient which multi- 
plied into the divisor will produce tho dividend. 

The number to be divived, is called the Dividend. % 

The number to divide byy is called the Divisor. 

The number of times the dividend contains the divisor, it 
called the Quotient. 

RULE. 

1. Assume as many figures on the left hand of the dividend as 
contain the divisor once or oftener ; find how many times they 
contain it, and place the answer on the right for the' first figure 
of the quotient. 

2. Multiply the divisor by the figure you have found, and 
place the product under that part of the dividend from which 
it was obtained. 

3. Subtract the product from the figures above it ; then bring 
down the next figure of the dividend and place it at the right 
handsof the remainder ; divide the number it makes up as be- 
fore, and proceed in this manner until the whole is finished. ' 



ARITHMETIC*, 



If 



How many times arc 5 
contained in 137906 ? 
PROOF. 

Multiply the divisor into the 
quotient, add the remainder if 
there be any, to the product ; 
if the work is right the sum 
will be equal to the dividend. 



&XAMFLE3, 

Dividend, Quotien 

Divisor 5) 137906 (27581 

> - 10 5 



37 proof. 137906 
35 

29 

25 



40 
40 

6 
5 



Divide 33489 by 9, 
Divisor, Dividend. 

9) 33489 (3721 Ana. 
27 9 

64 3S489 proof. 
63 

18 
18 



1 remainder. 
Divide 11680 by 32 
32)11680(365 Ana* 
96 

208 
192 

160 
160 



— * Proof by addition 1 1680 



9 
9 



Divide 1893312 by 912, Ana. 2076. 

Divide 1893312 by 2076, Ana. 912. 

How often does 761858465 contain 90001 ? Ana. 8465. 

Divide 280208122081 by 912314. Ana. 30714O TT Vrrr 
CONTRACTIONS. 

To divide when there are cyfihera at the right hand of the 
diviaor. 

RULE. 

Cut off the cyphers from the divisor, and just the same num- 
ber of digits from the right of the dividend ; then divide the 
remaining figures as usual, the quotient will be the answer.— 



* Add the remainder and all the products of the several quo- 
tient figures (multipled by the divisor) and the sum, if the w«(k 
be right, will be equal to the dividend. 
3 



IS 



VULGAR 



To the remainder (if there be any) annex those figures cut off 
from the dividend* and you will have the true remainder. 

EXAMPLES. 



(0 

Divide 460000 by 1200 
12(00)4600)00(383 Ana. 
36 


(2) 
Divide 7600 by 40 
4(0)760)0(190 Ana, 
4 


100 




36 


96 




36 


40 




a 


36 







400 true remainder. 

3. Divide 7380964 by 23000. 

4. Divide 11659112 by 890000. 
5 Divide 9187642 by 9170000. 
6. Divide 29628754963 by 35000. 



Ana. 390*9t& 
Ana. 131„iiw. 
Ana. l?l$Uhr 
Ana. 846535^f|4f 



SHORT DIVISION. 

RULE. 
Find how many times the divisor is contained in the fir* 
figure or figures of the dividend, place the result under, and 
carry as many tens to the next figure as there are ones over. 

EXAMPLES. 

Dividend (2) 

1. Divisor 3)76432* 4)1134152 



t 254776—1 rem. 


283538 


(3) 
5)649871923 


6)1027182341 






00 
')25OOti321702 


(6) 
8)11297653009 






)8701 256620 


10)1097654321 



ARITHMETICS 



11)3076259862 



(10) 
12)175634589 



I*. 



SECONDLY. 
When the divisor is a composite number, or the product of 
two or more numbers in the table. 

RULE. 
Divide successively by the component parts of the given 
divisor. 

EXAMPLES. 

1. Divide 9125 by 25. 

5X5=25; the component parts of the given divisor 
then, are 5 and 5. Thus, 

5)9125 

5)1825 









36: 


5 quotient \ 


2. Divide 


178464 


by 


16. 


Am. HI 54 


3. Divide 


79638 


by 


36.* 


Am. 2212^ 


4. Divide 


9S7387 


by 


54. 


Am. 17359^ 


5. Divide 


93975 


by 


84. 


Am. 1U8& 


6. Divide 


145260 


by 


108. 


Ana. 1 345 


7. Divide 


1575360 


by 


144. 


Am. 10940 



To Divide by 10, 100, 1000, 10000, &c. 
RULE. 
Cut off so many figures from the right ©f the dividend, a* 
there are cyphers in the divisor ;— that part cut off! from the 
dividend is the remainder, the other figures in the dividend are 
the quotient. 

EXAMPLES. 

(0 
Divide 600065 by 1000 

1(000)600)065(600 quotient, and 65 remainder. 

2. Qmde- -' 65 by 10. Am. 36 and 5 rem. 

3. Divide 5762 by 100. Am. 57 and 62 rem. 

4. Divide 90764 by 1000. Am. 90 and 764 rem. 

5. Divide 876432 by 10000. Am. 87 and 6432 rem. 



* The total remainder is found by multiplying the last re- 
mainder by the first divisor, and adding in the first remainder* 



30 VULGAR 



SUPPLEMENT TO MULTIPLICATION. 

To multiply by a mixed number, that is a whole number and 
a fraction. 

RULE. 

Multiply by the whole number, and take f , £, |, Sec. of the 
multiplicand, and add it to the product. 

EXAMPLES. 

1. Multiply 43 by 12$ 
32 



Ana. 5374 



. ( 2 ) 

Multiply |)24 
by Si 




■ ■ «» 




i)12=i 
6=4 
72 


Ana, 


90 




f 


An*. 


20533-f 


Ana* 


334134 


Ana. 


1191418 


Ana. 


6598| 



3. Multiply 2464 by 8£ 

4. Multiply 6497 by 5f 

5. Multiply 12248 by 9| 

6. Multiply 345 by 19-J. 



Practical Que at ions in Multiplication and Division. 

1. In 36 pieces of broad cloth each containing 24£ yds. 
how many yds. ? Ana. 873 

2. What is the product of 430 multiplied into itself ? 

Ana. 184900 

3. What number multiplied by 9 will make 225 ? 

Ana. 25 

4. What cost 9 yds. cloth at g7 pr. yd ? Ana. g 63 

5. If a man spend 8600 pr. year, what is that pr. calender 
month ? Ana. 8 50 

6. Sold a ships cargo for 87940, required to find 1-4 of the 
amount ? • Ana. 8 1985 

7. The quotient of a certain number is 1 1940, and the divisor 
20) What the dividend ? Ana. 238800 

8. How many feet are there in a mile, or 320 rods, allowing 
eath rod contains 16f feet ? Ana. 5280 

9. How many yards in a mile, if $* yds. make one rod ? 

Ana. 1760 

10. How many yards of broadcloth, at 87£ per. yd. may be 
• bought &r 37 yds. of do. at 8 S| per. yrd. ? Ana. 18£ yrds. 



ARITHMETICS 31 

1 1 . How much wine, at 1 J doll. pr. gal. with molasses, at £doU, 
pr. gal. and of each an equal quantity, must be had in exchange, 
for 35 gallons of brandy, atg 2,25, pr. gal. and 10 gallons of eld 
spirits, at $ 1,1 2$ pr. gal. ? Am. 40 gallons.* 



COMPOUND ADDITION. 

Teaches to unite several numbers of different denominations— 
as pounds, shllings, pence, &c. into one sum. 

RULE.* 

1. Place numbers of the same denomination under each 
other. 

2. Add the figures in the right hand column, and find how 
many of the next denomination are contained in the sum, 
which carry to the next denomination-, observing to set down 
the remainder under the column added, and thus proceed with 
all the columns excepting the last, where the whole sum is t» 
fee written down. 

STERLING MONEY. 

£ . *. d. . £. *. (/. yr. £. *. d. gr, 
17 13 4 84 17 5 2 47 13 6 O 



13 10 2 75 13 4 1 19 2 9 2 

10 17 3 51 17 8 3 14 10 11 1 

& 8 7 20 10 m 1 12 9 i 3 

3 3 4 17 15 4 2 8 7 6 2 

8 8 , 10 10 11 12 7 t 



54 



54 1 



261 


5 


8 


1 


176 


8 


2 


3 


261 


5 


8 


1 



* The reason of this rule will be obvious, if we consider, that 
1 in the column of pence, is equal to 4 in the column of forth* 
ings, and 1 in that of shillings; to 12 in the column of pence fee. 



V?- VULGAR 

•(*)• ■: w ■ (6) 

£. 9. d. gr. £ . *. rf. gr. £. 9. d. yiv 

II 9 6 1 144 9 12 I 987 6 9 1 

19 8 4 • '»• 160 19 10 3 17 19 11 2 

99 11 10 2 140 4 2 6 4 3 

6 O 4 1 910 4 3 89 6 10 2 



2. TROY WEIGHT. 



lb. 


or. /iwf. $>r. 


lb. 


OZn flXOt. 


gr. 


lb. 


oz. /l«tf . 


gr* 


19 


11 19 23 


11 


9 6 


4 


9 


11 19 


23 


Jl 


10 11 20 


20 


7 3 


20 


8 


10 18 


22 


6 


9 17 10 


10 


9 16 


17 


7 


9 17 


21 


4 


8 4 9 


9 


8 14 


23 


19 


8 16 


20 


3 


7 12 17 


17 


4 19 


3 


18 


7 15 


19 



3. AVOIRDUPOIS WEIGHT. 



wt. 


yf. 


/*. 


lb. 


oz. 


dr. 


r. 


C*tff. 


gr. 


lb. 


oz. 


2 


3 


27 


25 


13 


15 


90 


4 


2 


17 


14 


4 


2 


18 


24 


10 


14 


100 


.4 


3 


27' 


15 


6 


1 


17 


23 


11 


13 


86 


19 


2 


19 


12 


9 





16 


18 


9 


10 


H 


13 


1 





• 


3 


« 


14 


27 


7 


8 


96 


10 


3 


17 


14 



4. CLOTH MEASURE. 



Y<L 


gr. 


na. 


JS.£. 


gr. 


na. 


£.F. 


gr. 


na, 


70 


3 


3 


44 


2 


2 


90 


SL 


3 


13 


2 


1 


60 


1 


3 


108 


\ 


3 


90 








90 


1 


3 


76 


3 


2 


8 


1 


3 


30 


3 


3 


40 


1 


3 


6 


3 


1 


20 





3 


95 





2 



5. DRY measure: 

fik. gt. fit. bu. file. gt. bu. fik. gt. fit. 

17 70 3 4 95 2 1 1 



2 


6 


1 


60 


2 


1 


76 


2 . 


.7 





3 


5. 


1 


55 


3 


7 


40 


2 


6 


1 


1 


4 





60 


1 


6 


26 


2 


4 





2 





* 1 


9 


1 


6 


3 


2 


5 


I 



ARITHMETICS <%s 

9 WINE MEASURE. 



tal. 


f* 


fit. 


hhd. 


gal. 


gt. 


fit. 


fiiflc 


hatib 


psl 


39 


3 


1 


4* 


61 


3 


1 . 


34 


2 


2 


36 


3 





20 


35 


2 





11 


1 


3 


35 


2 


1 


ifr 


24 


1 


1 


7 


3 


2 


32 


3 





17 


U 


3 





19 


1 


3 


11 


2 


1 


10 


9 





1 


45 


1 


,1 



7. LQNG MEASURE. . 



Yds. 


ft. 


in. be. 


m. 


fur. 


fiol. 


le. m. 


fur. 


fioL 


4 


2 


11 2 


46 


3 


16 


85 2 


7 


27 


1 


2 


9 2 


91 


1 


29 


75 2 


5 


19 


2 


1 


8 1 


67 


3 


IS 


25 1 


4 


23 


1 





7 I 


60 


7 


33 


95 1 


6 


11 


3 


1 


10 1 


35 


2 


11 


11 2 


3 


15 




1 


*. LAND, 


OR SQUARE MEASURE 






teres 


roods 


rocf« 


acre* 


\ roods rods 


acre* 


roods 


rodi 


440 


3 


37 


11 


3. 


17 


990 


3 


39 


760 


2 


33 


97 


2 


16 


760 





14 


600 


1 


14 


20 


3 


la 


17 


3 


38 


976 


2 


35 


36 





24 


32 


2 


20 


57 





20 


25 





8 


203 


3 


34 



9. SOLID MEASURE. 



r. y*. 




cord*. 


ft. 






/'* 


inches-. 




40 90 




. 3 


127 






20 


1440 




203 7 




20 


220 






26 


1259 


•% 


23 23 




35 


29 






22 


1440 




__ — 




10. 


TIME. 










Yrs> mo. 


w- 


dye. 


; 


Yrs. 


<fy«. 


h. 


m. 


sec. 


57 7 


3 


26 




22 


300 


23 


59 


34 


230 6 


2 


5 




33 


327 


23 


44 


43 


3 10 


4 


30 




2*8" 


364 


20 


43 


58 


19 9 


3 


9 




34 


303 


23 


34 


33 



J 



94 



r 








VULGAR .. 








• 






11, 


CIRCULAR MOTION. 




# 




&. 


o 


\ 


W 


S. 


o 


.» \* 


S. « 


v . 


t* 


3 


29 


17 


14 


11 


29 


50 40 


17 


13 


50 


2 


23 


57 


44 


10 


20 


30 45 


34 25* 


49 


35 


4 


22 


20 


40 • 


37 


18 


48 29 


42 10. 


38 


r 


1 


30 


S3 


40 


23 


17 


59 57 


20 12 


36 


54 


4 


14 


45 


55 


99 


29 


40 50 


44 20 





33 


6 


24 


42 





8 


17 


39 42 


5 27 


15 


42 























COMPOUND SUBTRACTION, 

Teaches to find the difference between any two sums of dif- 
ferent denominations. 

RULE. 

Place numbers of the same denomination under each other, 
the less below the greater ; begin with the least denomination, 
and if it exceed the figure^ above it, borrow as many units as 
make one of the next greater ; subtract it therefrom, add the up- 
per figure to the difference ; always adding one to the next 
higher denomination for that which you borrowed. JPjROOi^-— 
the same as in Simple Subtraction. 

1. STERLING MONEY. 

EXAMPLES. 

£ . *. d. qr. £. s, d. qt. 

From 346 16 5 3 494 17 9 3 

Take 128 17 4 2 479 19 10 2 



Bifference 217 1? 1 1 



lb. oz. fiwt. 

12 10 19 

9 11 18 



2. TROY WEIGHT. 

oz fiut. gr* 

10 19 23 

6 17 19 



lb. oz. fiwt. gr. 
120 10 16 19 
134 11 15 17 



arithmetics:. js 

8. avoirdupois weight- 

ib. *z. lit. twt. qr. lb. T. cvt. qt. fa ox. <Jr. 

9 r 15 7 3 13 3 10 3 54 IS 1$ 

4 9 26 5 2 15 2 12 2 29 14 15 





74. 

13 


3 $ 
% I 






tfik. qt. 

1 1 

2 




?,al. 
39 
36 


?'. /^r. gi. 

3 12 
3 O 





4. CLOTH MEASURE. 

JB.Ji.-fr. na. E.JP. qr. «*« 

64 2 2 190 2 B 

40 13 08 1 3 



8. DRY MEASURE. 

bu. fyk. qt. bu. ilk. qU 

70 3 4 95 2 1 

60 2 1 76 2 7 



6 WINE MEASURE. 

A&eL g?L $$* jOr hhd. gal. qt. fit, 
261 3 12 34 2 2 1 



.- 1 ,■ 1 'U.H J f .M ■ I l lu ll. W l l ■ — IMp, ' 

Yds. ft. in. be. m. fur* flol. le. m* fur. fiol* 

,4 %, U ,2 -.. ?ft $, - i$., . S5 2 7 27 

' 1" %; . 9 V 2 \ , 4\ ..■ i . 2* . . 75. ,,$ * • 19 - 



*. LAN©, OR SQUARE MEASURE. 

vfcri* r**rf* ro<& , ech;* rood* rorft \feet. inches 
760 2 38 97 2 16 960 14 

440 3 .37 ... H 3 17 790 33 



m 



VULGAfc 



V. SOLID MEASURE. 

3P. ft* <#rd%. ft. r T. ft., inched 

240 90 3 127 1440 20 238 

03 7 2 220 ' 26 18 135 



fr*. 

230 
57 



7 

>6 



3 26 
~2 5 



10. TIME. 

Yt*. dya. h. m; sec. 

32 300 2S 59 34 

23 327 23 44 43 



11. CIRCULAR MOTION. 



3 29 17 14 
2 ?3 57 44 



H 29 50 40 

10 20 30 45 



S - o \ M 

37 13 69 
14 25 49 35 



* . - 



COMPOUND MULTIPLICATION. 

Shows how to find the amount of any given number of diverse 
denominations, by repeating it any proposed number of times. 
FEDERAL MONEY. 

-;, RULE. 

Multiply as in whole numbers ; and place the *eperatrix as 
many figures from the right hand in the product, as it is in the 
multiplicand. v ~ 

EXAMPLES. 



8 cts. 
Multiply 17 18 by 25 
25 



>h G. C. I 



doh It. c. In. 

Multiply 7 13 9 

14 



by 14 



85 90 

343 6 



Preduct. g 428, 50 



> 28 5 5 6 
71 3 9 

Product. $ 99, 9 4 6 



3. Multiply 1 1 mills by 40 Ana. 44 

4. Multiply 41 cents 5 mills, by 150 Am. 62 25 



ARITHMETKX. "& 

S. Multiply * dols. by 50 fna* 450 00 

^ 6. The number of inhabitants in the United States is 7 mil* 
lions ; if each should pay the sum of 8 eents yearly, for 9 year*, 
how many dollars would be raised I Ana. 5 millions. 



WEIGHT^ MEASURE, STERLIJYG MOJVEYy&c. 

RULE. 

Write the multiplicand, and place the quantity under. the least 
denomination for the multiplier? observe- the same rules, for car- 
lying as in compound addition. 

1. What cost 9 lb. of* sugar* at 2*t 8<& 2gr. pr.lb, I 
2* 8$rf , 



Ana. XL 4* 4%d 



2. 3 lb. of green tea, at 9*. 6d. pr. lb. ? An*. £ \ , 8* Sd 

3. $ lb. of loaf sugar> as 1 *. . 3d. pr, lb r ? Ana. £0 '6,3 
4. 9 cwt. of cheese, at 1/. 11*. Sd. pr. cwt. Ana. 14/ 2a 9d 

When the multiplier excteda 12.; 
Multiply successively by its component parts* instead of the 
whole number. 

Examples. 
L 16 cwt of sugar, at XL 18*. 8d. pr. cwt. 
(1) (2) 

1/ 18* %d 28 yds.. of broad cloth, at 19*. 4d. 

4 J»r.yd. Ana. £27 ', Xa.4d. 

7 14 8 

4 



£ 30 18* Sd 



3. 182 yds. Irish linen, at 2*. 4d. pr. yd. v Ana. #15, *» 
*4w 144 reams of paper, at 13*. 4d. pr. ream. Ana. £ 96, 
5.^96 bushels wheat, at l/. 3*. 4d. pr. bush. Ana. £ 112, 

If no two numbers multiplied together will exactly equal the 
multiplier, multiply by any two numbers that come the nearest, 
then multiply the upper line by the remainder, which added to 
the last product gives the answer. 



f VULGAR 

I. What will 47 yds* hroad cloth come to, at Its. *<t. pr.jd. ? 
£ ** & 
0, 17, 9 price of 1 yd. 



5 



4 


8 


9 
9 


39 


18 


9 


1 


15 


6, 


41, 


14 


3 



price of 5 yds. 



price Of 45 ft* 
price of 2 yds. 

Ana. 41, 14 3 " price of 47 yards* 

2. 29 yds.of cambricd,at 0/. 13*. ?d. pr. yd. 

wfiu.i"19, 13*. HA 
3* 111 yds. broad cloth, at 1/. 2*. 6^. pr. yd. . .-. 

Ant.£\ < Uy 17*. 6rf. 
4, 23 ells, at X>£ 3*. 6ftf. pr. ell ? 4n*.£4> U* sd.\ 

8. 117 cwu Malaga raisins, at £ly 2*. 3rf. pr. cwt. 

An*.£lj>0 9 $a* 3<f. 

6. 59 yds., tabby velvet, at 7s. lOd. pr. yd. Arts. £ 23, 2*. 2tf. 

7. What is the weight of 7 hhds. of sugar, each weighing 
9 cWt. 3 qr. 12 lb. I An*. 69 cwt. 

8. In 9 fields, each containing 14 Acres, 1 rood, and 25 pole% 
bow many acres \ Ana. 1 29a. iqr. 25rods. 

9. In 6 parcels of wood, each containing 5 cords and 96 feet, 
bow many cords ? Ana. 34-J cords. 

& * k \\ . " &• ° v vv 

10. Multiply 1 15 48 24 by 5. Arts. 7 19 2 O 
ill. Multiply 3 cords, 87 feet, by $ Ahs* 29 cords 56 feet. 



COMPOUND DIVISION. 

Teaehes to find how often one number may be had in another 
of different denominations. 

ROLE* 

Write down the given sum in cents, and divide as in %hele 
lumbers ;— the qHotient will be the answer in cents. 



* Note. If the cents in the given sum are less than l6, place 
a cypher on their left, or in the ten's place. 



ARITHMETIC*. %? 

EXAMPLES. 

Divide 8 674 19 cents by 24 
(1) dol. cts. m. 
24}67419(28, 0$ 1J 
48 

To bring cents into 1 94 Wh,en there is a remainder, 

dollars, you need only 192 add a cypher, again divide* 

point off 2 figures on - — and you will have the mills, 

the right, the rest 219 

will be dollars. 216 

3a 

24 



2' Divide 4 dols. 9 cents, or 409 cents by 6. Ana. ,68, cts* 

3- Divide 9 dols. 24 cents, by 12 % Ana. ,77 cts. 

4 Divide 2 dollars into 33 equal parts/ Ana; ,66, cts, 

5. Divide 1000 cents by 25. Ana. j40 cts. 

6. Divide 999 cents; by 9. Ana. Ill cts. 

7. Divide 1 <!<>!. by S mills. Ana. ,500 mill* 

8. Divide 125 dols. by 500. Ana. 25 cts; 

9. Divide 10 cents, by 10 mills* Ana. 10 mills. 

STERLING MONEY, WEIGHTS, MEASURES, &c. 

UULE. 

Begin at the left hand, as in simple division, and if any thing 
remains, determine how many of the next denomination the .re- 
mainder is equal to, which add to the next denomination, con- 
tinuing to divide, and to carry the remainder, as before, till the 
whole is finished. (1) 

£. a. 'd. M 
Pivide 19 4 10 3 by 4, 



£. 


8. 


d. 


?r, 


19 


4 


10 


3 


4)19 


4 


10 


3 



£.4 16 2 2 | Ana. 

£. a. d. 9 r. 

2. Divide 31/. 14*. 9d. 2gr. by 17. Ana. 117 4 

3. Divide 119/. 12*. 2d. 3gr. by 9. Ana. 13 5 9 2 

4. Divide 1/. 19*. Bd. Ogr. by 1 1. Ana. 3 7 f 



Note. When the divisor exceeds 13, and is the product of 2 
numbers ; divide by one of these numbers first* and the quotient 
by the other, the last quotient will be the answer. 



30 VULGAR 

5. Divide 128*. Sii.Od. Oqr. by 42'. Am. 3 S 1 Q !► 

6. Divide 5/. 10*. id. Oqr. by 81. Am. 14 £ 

7. Divide 6 tons, 1 1 cwt 5 q r »- 19jl >- bv *• 

^>w # I r. 12 rwf. Syn 25#. ISoau 

8. A. piece of cloth containing 24 yds. cost £. 18. 6*. what is. 
it pr. yd. ? Ans* 15*. 3d 

9. Divide IT Ida. 1 m. 4 far. 21 pols. by 21.. 

Am. 2 m. 4 fur. 1 pol. 

10. From a £iece of cloth containing 64 yds. and 2 nails, a. 
aylor was directed to make 27 coats ; what did each coat coHr* 

lain f Ana. 2 yds. 1 qr. 2 na. 



REDUCTION. 

Teaches to change numbers from one denomination to aneth- 
<sr, without altering their vahie. 

Reduction is either Ascending, or Descending. It is Ascen- 
ding, when numbers of a lower denomination are raised to a~ 
higher denomination. It is Descending, when numbers of a 
higher denomination are reduced to a lower denomination. 

PROOF. Invert the order of the question. 
REDUCTION DESCENDING. 

RULE. 

Multiply the highest denomination given, by so many 'oS the 
next less, as make one of that greater ; and thus continue til£ 
you have reduced it as low as the question requires. 

EXAMPLES. 

!. In 37/. 13*. Id Zqr. hew many farthings. I 
20 

753 Shillings.* 4)3617* 

12 — 



$043 pence. PROOF* 



12)9043 -f 3qr r 



I 20)75,3 -f- 7d. 

As. 36175 farthings^ £37,13,7,3 



2. In 23/ lit* 7\i. how many farthings ? Am. 22638. 

3. In 47/. 19*. 3d. how many shillings, pence and farthings £ 

Am. 959*. 1 15 lid and 46044?r. 



♦Note. In multiplying by 20, add in the shillings, by 12 add 
ia the pence, and by 4, add in the farthings, if any, in all similar 

&S8ft. 



ARITHMETIC*. *l 

4. In 315 dolls. 50 cenu, how -many threepences and farth* 
Ings ? ./fa*. 7572 threepences, 90864 qr. 

5. In 121 French crpwns* at 6*. 8c?. each, how many pence and 
ifcrthings? An*. 9680d. 38720 qr. 

6. In 312/8*. 6id. how mariy sixpences and balf-pences ? 

An*. 12497 sixpence*, 149964 half pences. 

REDUCTION ASCENDING. 

RULE. 

Divide the lowest denomination given, by se many of that de- 
nomination, as make one of the next higher, and so on through 
all the denominations, as far as the question requires. 

FRO OF. Multiply inversely by the several divisors, 
examples: ;, 

1. In 122318 farthings, how many pence, shillings, and 
founds ? 

Farthings in a penny = 4) 1 223 1 8. 

Perice in a shilling =» ; 12)30579 -f- 2qr. 

. Shillings in a pound =: 2,0)254,8 -J- 3d, 

An*. £127,8s.3d.2qrs. 

2. In 30329 farthings, how many pounds \ , , . 

AH*. £ n t lU lOrf. Iqf. 

3. In 46044 farthings, hofr many pence, shillings, and pounds? 
- ■' AH*}t\$il&. 9594.47*. 

r 4. In 90864 farthings, how many dollars ? Ana. % 315*. 
• 5. la 20160 pence, how many pounds ? An*. £ 84 ' 

2. TROY WEIGHT. 

1. In a dozen of silver spoons weighing lib. 3oz. 1 Ipwt. how. 
many grains ? Ana. 7464 grains. 

2. In 10 ingots df gold,«ach weighing 9oz. 5pwt. how many 
grains? An*. 44400 

. 3. How many table spoons weighing 23pwt. each, and tea 
spoons 4pwt« 6 grs. each, an4 an equal number of each sort, can 
be made from 41b. loz. ipwt. of silver. An*. 36 A ; 

REDUCTION ASCENDING AND DESCENDING. 

3. AVOIRDUPOIS WEIGHT. 

1. In 19 lb. 14oz. 1 1 dr. how many drams ?- -Ana.^ 5099 

3. In 5 tons, how many drams? Am. 3867200 

3. Bring 5099 drams into pounds. An*. • 19/a, 14oz. 11 dr. 

4. A merchant has 5 hhds. of tobacco, each 8 cwt. 3 qrs. 141b. 
and wishes to put it into boxes containing 701b. each, how many 
bcrxes are requisite ? Ans r 71 



3* VULGAR 

-■$: CLOTH MEASURE. 
1< In 5469 nails, how many yards ? Ans. 341 yds. $ qr. \ ttel 
• *. In 1320 nails, how many Ells English i Ans. 61* 

3. In 28 Ells Flemish^ how many quarters ajDd nails ? 

•in*. 84 qr. 336 na. 

4. How: many coats containing 1| yds. each, can he made 
from 75 J- yards of broad cloth ? Ans* 42* 

5. DRY MEASURE. 

1. In 63 bushels, how many peaks, quarts, and pints ? \ 

Ans. 272 pecks, 2176 qts. 4352 pt$» : 

2. In 25 bush. 3 pks. 7qts. how many quarts ? Ans.'szi* 

3. In 8704 pints, how many bushels ? Am. 136. 

4. A gentleman has 1003 bush. 3 pks. of grain, and a teayra 
that consumes 2| bu^h. pr. day, how long will the grain last ? 

An*, one jre^r 

9. WINE MEASURE, . , , > 

1. In 9* tons of wine, how many quarts ? Ans. 9072. 

2. In 18144 pints of wine, hpw many hhds.? An ft* 56. 

3. How many bottles containing If pint, can be filled from a 
pipe of wine? ^ Ans. 672. 

r. LONG MEASURE. ; r . 

1, In 1 7. miles, how many inches ? Ans. 1077120. 

X In £. leagues, how many yards? An*. ls5R4Q« ? 

3. How, many revolutions do the forward wheels of a stage 
describe in running from Hartford to New-Heven* it bfipg ?4i 
miles, allowing the wheels to be 14| feet in t circumference ?. 

x Ans. 12114VVT 

4. What is the circumference of the globe in inches, it being 
$60 degrees? Ana* 1 $«$?#• ?<>9 

M ' '.'.-. /■ !'■ 8. TIME. . ... ....,., 

i. In 1 year, or 363d. 5h. 48^ 58>\jiow many second* ? 4 >i .^ 

Ans. 3155MW 
%- In 655989 days, bow many years, reqHonia^ th* ,y$ajr to 
contain 365d. 6h« ? A»*i* \%$& 

' 3. How many minutes were there from the fcrith frfCtate Un' 
the year 1776, allowing the length of the year the same as in the 

first example? ■•••../- Aas. ?34p$5364 v 4$* 

....■*'• ■ ■ - . i * •■.*'.» ■ * ' * • • • * * * ■ ..* 

9. CIRCULAR MOTION, f: 7 \ 

1. In 9 signs, 13° 25* how many seconds? Ans. 1Q20PQQ 

2. In 811480 seconds, how many signs? 

Ans, 7§. U o M\40W 

3. How many minutes in the whole of th* Earth's orbit, or 
12 signs ? An?. 4 1.60^ 



ARITHMETIC*. S3 

N 

N 

OF CURRENCIES. 

HULR* 

Divide the given sum reduced to shillings; to sixpences, or 
pence , by the number of shillings, or pence, in a dollar, in 
each state. 

1. Reduce £6S 15s. New England currency to Federal Moi\* 
ey. Ans. g 212 50 cts. 

2. Reduce £"481 New York currency to Federal Money. 

Ans. g 1077 50 

3. Reduce ^"37 10s. Pennsylvania currency to Federal Mon- 
ey. Ans. g 100. 

To change Federal Money to the currency of each state. 

RULE. 

Multiply the given sum in cents, by the number of pence in a 
dollar, and cut off two figures to the right of the product, what 
is left will be the answer in pence, and if the figures thus sepe- 
rated, be multiplied by 4 ; and 2 figures again cut off as before, 
those at the left hand will be farthings. 

EXAMPLES. 

Reduce g 438 42 cts. to New- York currency. 

Ans. £\75 7s. 4£d^ 
Reduce g 1971 96 cts. to New-England currency. 

' Ans. £591 lis. 9d. 

Reduce 85 dolls. 43 cts. to sterling money. 

Ans. £19 4s. ad. 

Note — When the given sum is dollars, multiply by the nuitt* 
her of shillings in a dollar. 



54 VULGAR 



«j ?. •sOTr» OOC0OKO " N oow 4. 

2J S £ I .oo«o*o^.o © a m o T 

•2 -S 5 -suwol « 000 > c> ' 0,£ > ,s < ~ o <* <* £ 

£ £> -s^noai-fiS^S^^^^^ --oof 

g South-Carolina 'p'ooooo.«oo« 

•O and 

(£i Georgia. 



f^00©00>*00<© 
% ^ Q O O » — — 00 K 




§ New-Jersey, Delaware rtioo<ooo<ooo co <© co to ^ 

to and 4 »ao2«22»'N « ** - - A 

S Maryland. l^o « ^ w - -< - - ooooT 

g 1 I • ,i, mm _ ^ 

52 XT— « V~«U /"^lOO. O O Q O O O ft O rt» K. T 



55 



New-York 
and 
•g. South-Carolina, 

•12 



r^oooooooo o»oo»**-T 

J (q00 ^ <OC0N «A 00 00 oo ^ ^ A 
t^)<0 Wujn-*hh^ OOO O JL 



~ Vermont, New-Hamp- p*ooooo<oo.o «©*<"£ 

.ti shire, Massachusetts, \ .<ococo«>co**c*c* <o to - ~ ¥ 

£ Rhade-Islatid Coanecti- ] - ^ ^ 

*cut and Virginia. (Js*-* $c *•-• — — — : r« oooof 

SB i .. i . . . i ■ - — — t 

|j . Sterling money ^oooooooo o <o o o ^ 



T3 



of 



«> Great-Britain. 



l ^SS^^^^SIS .vj ^. -« o- A 
*-*s*co — o — -« — o o o o o o ±: 



52 Standing weights. - S«£coo»«oo»o»«m««p ooseowt 
a U— ~ —H—l 

§ * s s| « i 

3 .• -S -8.9 So a 

**« S £BQS meet* ^Q c 0h 



ARITHMETIC*. SS 



DUODECIMALS. 

Duodecimals are so called, because they decrease by twelves,. 
from the place of feet towards the right. Inches are sometimes 
called firimesy and marked thus * ; the next division, after inches, 
is called parts, or seconds^ and is marked thus " ; the next is 
third*) marked thus "' ; 8cc. 

Multi/ilicatien of Duodecimals ; or Cross Multiplication. 
RULE. 

1 . Under the multiplicand write the same denominations of 
the multiplier, that is, feet under feet, inches under inches, Ssc. 

2. Multiply each term in the multiplicand, beginning at the 
lowest, by the feet in the multiplier, and write each result un- 
der its respective term, observing to carry an unite for every 1 2 y 
from each lower denomination, to its next superiour. 

3. In the same manner, multiply every term in the multipli- 
cand by the inches in the multiplier, and set the lesultof each 
term one place farther toward the right of those in the multi- 
plicand. 

4. Proceed in like irmnner with the seconds and all the rest of 
the denominations, if there be any more ; and the sum of all the 
several products, will be the product required. 

The products of the several denominations depend upon the 
principle, that— 
Feet by feet give feet. Primes by primes give seconds. 

Feet by primes give primes. Primes by seconds give thirds. 
Feet by seconds give seconds. Primes by thirds give fourths. 
&c. Sec. 

Seconds by seconds give fourths. Thirds by thirds give sixths. 
Seconds by thirds give fifths. Thirds by fourths give sevenths. 
Seconds by fourths give sixths. Thirds by fifths give eights. 

&c. Sec. 

Or, in general 

When feet are concerned, the product will be of the same de- 
nomination as the term by which the feet are multiplied. When 
.feet are not concerned, the name of the product will be express- 
ed by the *um of the strokes, or 7:iarks over both the factors. 



3* VULGAR 

EXAMPLES. 

I. Required the contents of a ceiling 10ft- 4* 5" by 7ft. 8* 6' 3 

ft. * " 

10 4 5 

7 8 6 



72 6 11 
6 10 11 4 
5 2 2 « 



Feet 79 IT 0" 6 m 6"" Ans. 



2. How many square feet in a board 17 feet 7 inches long, 
and 1 foot 5 inches wide? ^ Ans. 24f. 10' 11" 

3. How many solid feet in a stick of timber 12 feet 10 inches 
long, 1 foot 7 inches wide, and 1 foot 9 inches thick ? 

Ans. 35f. 6' 8" 6*" 

4. Required the number of solid feet in a load of wood, that 
is 9$ feet long, 3-} feet wide, and 3 feet 7 inches high I 

Ans. 1 13f. 5* 8" 

5. How many yards of painting in a room 20 feet in length, 
14t feet in breadth, and lOjfeet in height, deducting a fire place 
of 4f. by 4^-f. and 2 windows, each 6f. by 3£f. ? Ans. 73 T * T yards. 

* In like manner may pounds, shillings, pence, 8cc. be multl* 
plied into each other, by observing the following principle—- 
that Pounds by pounds give pounds. 

Pounds by shillings give shillings. 
Pounds by pence give pence* 
8cc. 
Shillings by shillings, every 20 is 1 shilling, every 5 is Sd. 
and each 1 is 2 farthings, and 4 tenths of a farthing. 

Shillings by pence, every 5 is a farthing* and each one 2 
tenths of a farthing, &c. 

Pence by pence, every 60 is a farthing, and every 6 one tenth 
ef a farthing. 

^ EXAMPtES. 

1. Let it be required to multiply 31. 5s. 6d. by 21. 42s 9d* 
£ a. rf, 
3 5 6 

2 12 9 . ^ 

6 110 qrs. tenths. 
1 19 3 2 4 

2 5 19 ' 



Ans. Sl.ias.Qd.Oqr,^ 



ARITHMETICS Sf 

2. Let it be required to multiply 2s. 6d. by itself. 

^ Ans. 3d. 3qta.* 

In the above example, a fiound was considered the integer ; 
but when a shilling is taken for the integer, observe the follow- 
ing precepts ; — namely, that 

Shillings by shillings give shillings. 
Shillings by pence give pence. 
Shillings by farthings give farthings, Sec. 
Pence by pence, every 1 2 Is a penny, and each 3 a farthing. 
Pence by farthings, every 12 is a farthing, and each 3 is £ of a 
farthing, &c. 
Farthings by farthings, each J 2 is £ of a far thin gr 

examples 
1. Let it be required to multiply 2s. 6d. by 2s. 6d. one 
shilling being taken for the integer. 

s. d» (2) 

2 6 Multiply 2\ feet by 2* feet. 

2 6 2f. 6' 

2 6 

5 

1 3 o 5 

— — 13 



£0 6 



Feet 6 3' Ans. 

Where it is apparent, that if, instead of shillings, pence and 
farthings, we reckon feet inches and quarters, the result will 
be the same. 

Xhe two following questions are Sexcessimals. 

3. If two places differ in longitude 2° 12' j what is their dif- 
ference of time ? 

, Mult. 2° 12* 00". 00"' . 

by S* 59" 20"* the time in which the Sun passes 

• — — — through one degree. 

Ans. 8* 46" 32'" 

4. Two places differ in longitude 31° 27' 30", what is the 
difference in time of the Sun's coming to the meridian of those 
places, the Sun passing through 15° in an hour I 

31° 27* 30" 

4* 00" In 4 minutes of a solar day, the Sun 

passses 1 degree. 

Ans. 1° 6* 30" 00"* 



"* Note. Whence it is manifest that fractions multiplied, 
become less, in the same proportion as integers^ by multiplying, 
become greater. 



■'•3* VULGAR 



SIMPLE PROPORTION, OR RULE OF THREE. 

Teaches, that by having three proportionals given, to find a 
fourth, which multiplied into the first, shall be equal to the pro* 
duct of the other two: or the rule of three teaches, by- 
having three numbers given, to find a fourth, which shall have 
to the second the same ratio, that the third has to the first. 

Proportion in common Arithmetics, is generally consid- 
ered direct, or inverse. It is direct, when more requires more, 
or less requires less, and inverse, when more requires less, and 
less requires more. 

1. Observe that two of the given numbers in the question are 
always of the same kind, one of which must be the first number 
in stating, the other the third ; consequently that number which 
is of the same kind with the answer or thing sought, will always 
be the second, or middle term in stating. 

2. Observe farther ; the third term is always a demand, and 
may be known by its asking the question. 

RULE. 

1 . Write the numbers so that the term which asks the ques- 
tion may possess the third place, and that which is of the same 
kind with it, the first place, the remaining term will posses 
the second place. 

2. Bring the first and third terms to the same denomination, 
and reduce the second, to the lowest name mentioned in it. 

3. Multiply the second and third terms together, and divide 
jtheir product by the first, the quotient will be the answer in the 
same denomination as the second term.* ' 



* This rule is founded on the principle that if four numbers 
be proportional, the product x>( the extremes, is equal to the 
product of the means. 

Thus 4: 8:; 16:52, here 4X32 = 128 ; and 8 X 16= 128 : 
whence it is evident, that dividing the product of the means^ or 
the product of the extremes by the Jirst extreme, the other ex- 
treme is obtained. Thus— dividing 128 the product of the 
means, by 4, gives 32 for the other extreme ; and dividing the 
product of the meuns, or the product of the extremes by one of 
the mean*, the other mean is obtained. Thus— -dividing 128 the 
product of the extremes, by 8, gives 16 for the ether mean ; 
hence the propriety of the rule, in multiplying together the 2d 
and 3d terms, or the means, and dividing by the first term, or the 
first extreme, to obtain the other ; for it has been demonstrated, 
that the product of the means, divided by the first * extreme* 
gives the other extreme. 



ARITHMETICS 3* 

This- rule is applicable when the proportion is direct ; but 
when the proportion is inverse j that is, when the conditions ef 
the question require the answer to be greater, or less than the 
second term : Multiply the first and second terms together, and 
divide the product by the third. The quotient will be the an- 
swer in the same denomination as the middle term. 

EXAMPLES. 

1. If 6 yards cost 8 18) what will 12 yards cost at the same 
rate? 6 s 18 : : 12 

13 



6) 216 (36 dollars. 
18 

36 
36 

do 
If 1 8 dollars buy 6 yards, how many yards will 16 dols. buy ? 
18 ; 6 : : 36 (3) 

6 If 8 36 buy 12 yds. how many 

—~ will 18 dollars buy. 

18)216(12 Ans. S6 ; 12 : : 18 
18 " 12 



36 56)216 (6 Ans; 

36 216 

ae ooo 



40 VULGAR 

i 

4. If 12 gallons brandy cost 25 dols. 44 cents, what will be the 
price of 2 pipes, at the same rate. 

Gal. g Pipes., 

12 : 25,44 i : 2 

2 hhds. in 1 pipe. 



As. 12 







4 
63 


gal. in 1 hhd. 






12 

24 




; 2544 : 
252 


i 


252 


do. in 2 pipes 


5088 
12720 
5088 









12)631088 (525 90 6 J Ans. . five hundred and 

60 twenty-five dollars, ninety 

~ cents, 6 mills and two 

31 thirds, 
24 

"to 

60 

108 
lo8 



80 

72 



. * 

12 

5. If 5 horses eat 10 bushels of corn in a week, how many will 
35 eat in the same time ? Ans. 70 

6. If an ounce of silver is worth 90 cents, what is a cup 
worth that weighs 21b. 10 oz. Ans. g 37 80 

7. If 15° of the equator revolve through the meridian in one 
hour, in what time will 150° 51* 15" revolve through ? 

Ans. lOh. 15* 25° 

8. What is the tax upon g 50 97, at ten cents on the dollar. 

Ans.- g 509 70. 

9. What wilj 4 casks of raisins, weighing 2 cwt^ 2 qr. 25 lb. 



ARITHMETICS 41 

come to, at 16 cents pr. lb. ? Ana. £ 195 29 

10. *At the rate of 15 deg. pf. hour, how much of the equator 
will revolve through the meridian in 12 hours 2 min. *6 sec. ? 

Aps. 179 deg. 36 min. 30 sec. 

1 1. When the Sun is on the meridian of London, what o'clock 
is it at Mexico North America, 100 degrees 5 min. 45 seconds. I 

Ans. 5 o'clock, 19 ip. 37 sec. a. m. 

12. What o*clock is it at Moscow 37 deg. 45 min. east long % 
when it is noon at London ? An§. 2 o'clock 31m« p. m. 

13 If the Sun comes to the meridian of London, 4h. 45m. 
30 sec. sooner than it does at the Meridian of Cambridge, what 
is the longitude of Cambridge ? Ans. 71 deg. 20 m. w. 

14. suppose a Gentleman has an income of 8 1940 a year, 
and he spends 3 dols. 46 cents pr. day, how much will he have 
saved, at the years end ? Ans. 8 683 10 * 

15. Sound uninterrupted, moves about 1 142 feet in a second, 
1aow long then, after firing a cannon at Springfield before it will 
be heard at Hartford, it being 26 miles ? Ans. 2 m.O sec.-f^T 

16. In a thunder storm it was observed, that it was 6 seconds 
between the lightning and thunder, at what distance was the 
explosion ? And. 6S52 ft. = J^mile. 

17. Suppose a rocket was seen at the instant of discharge, i2 
seconds before the report, at what distance was the gun. 

Ans. 2{|£ miles. 

18. If 8 100 in one year gain 8 6> what will 8 314 15 cts. 
gain in the same time ? Ans. 8 18 84c. 9m. 

19. If 8 212 25 c. gain 8 12 37* in one year what is that pr. 
cent ? Ans. 6 

20. A owes B 8 1?36 59 cts. but becoming a bankrupt, he 
is unable to pay more than 65 cents on the dollar, what does B 
receive for the debt ? Ans. 8 H28 73c. 3^rm 

*fel. If a man buy merchandize to the amount of 8 560, and 
gain by the sale 8 190 40, how much will he gain by laying 
out 150 at the same rate ? Ans. 8 50 00 

22. If 30 men perform a piece of work in 1 1 days, how many 
. men can accomplish another piece of work 4 times as large in, 

a fifth part of the time ? Ans. 600 

23. A wall that is to be built to the height of 27 feet, was rais- 
ed 9 feet by 12 men in 6 days, how many men must be employ- 
ed to finish the wall in 4 days, Working at the same rate ? 

Ans. 36 

24. If a stick 8 feet long, cast a shadow on level ground 12 



* Note. The equator may always be supposed to revolvo 
through the meridian, at the rate of 15 degrees in 1 hour of so- 
lar time, without any sensible errour ; though it is a f ruction 
wide of the tiuth. 

6 



42 VULGAR 

feet, what is the width of a river, over which a tower, known te 
be ISO feet in height casts its shade. Ans, 270 feet. 

OF THE LEVER OR STEELYARD. 

It is a principle in Mechanic ks, that the power is to the 
weighty as the velocity of the weight, to the velocity of the pow- 
er ; therefore to find what weight may be raised or balanced by 
any given power, say ; 

As the distance between the body to be raised, or balanced 
and the fulcrum, or prop, is to the distance between the prop 
and the point where the power is applied ; so is the power to 
the weight which it will balance. 

If a man weighing 1601b. rest on the end of a lever 10 feet 
long, what weight will he balance on the other end, supposing 
the prop 1 foot from the weight I 

The distance between the weight and the prop being 1 foot, 
the distance from the prop to the power is 10*-* 1 = 9feet > - 
/ therefore, 

Je. ft. Ih. lb. 
As 1:9 : : 160 i 1440 Ans. 

If a weight of 1440 be placed 1 foot from the prop, at irhat 
distance from the prop must a power of 1601b. be applied to. 
balance it ? 

As 160 : 1440 : i 1 : 9 feet. Ans. 

At what distance from a weight of 14401b. must a prop be 
placed, so that a power of 1601b. applied 9 feet from the prop, 
may balance it. 

As 1440 : 160 : : 9 : 1 ft. Ans. 

The celebrated Archimedes said he could move the Earth, 
if he had a place at distance from it to stand upon, to manage his 
machinery. % t 

Now suppose the Earth to contain in round numbers 
4,000,000,000,000,000,000,00000 lb. or 400000 Trillions of lbs* 
and that Archimedes was suspended from the end of a lever 
12,000,000.000,000,000,000,006,000 miles in length, and the 
fulcrum, or centre of motion of the lever to be 6000 miles from 
the Earths centre, how much must Archimedes weigh to balance 
the Earth * Ans. 200 lb. 

OF THE WHEEL AMD AXLE. 

The proportion of the wheel and axle, (where the power is ap- 
plied to the circumference of the wheel, and the weight to be 
raised is suspended by a cord, which coyls about the axle as the 
wheel turns round,) is as the diameter of the axle to the diatnc- 



ARITHMETIC*. 43 

terofthe wheel, so is the power applied Co the wheel) to the 
weight suspended from the axle. 

Suppose a windless is constructed in such a manner, that 14lb* 
applied to the wheel will raise ?24lb. suspended from the axle, 
which is 6 inches in diameter, what is the diameter of the 
wheel ? Ans. 8 feet. 

lb. in. lb. in. 

Inversely 9 As 224 : 6 : : 14 : 96 = 8 feet. 

Suppose the diameter of the wheel to be 8 feet,required the 
diameter of the axle, so that 141b. suspended from the wheel, 
may balance 2241b. on the axle. 

lb. in lb. in. 

Inversely, As 14 : 96 : t 224 : 6 diameter required. 

Suppose the diameter of the wheel 96 in. and that of the axle 
6 in; what weight suspended from tbe axle will balance 14ltu 
upon the wheel ? 

Inversely, As 96 : 14 : : 6 t 224 weight required* 






OF LOGARITHMS. 

THE operations of Multiplication and Division, when they 
are to be often repeated, and the extracting of Roots, especially 
if they be from the higher powers, become so tedious,, that it is 
an object which has long employed the skill, and talents of the 
roost profound mathematicians, to substitute in their place 
jnorc expeditious, and easier methods of calculation.— -To effect 
this, certain numbers have been so contrived, and adapted 
to other numbers, thajt the addition and subtraction of the former, 
fcave been made to perform the office of multiplication and divU 
sion in the latter, with imcomparable facility find expedition. 

The invention of Logarithms is by some ascribed to Baron 
Napier. But the kind of Logarithms now in use, was invented 
by Mr. Henry Brirggs, Professor of Geometry in Gresham Col* 
lege, London. 

LOGARITHMS (from logos, ratio and arithmos* number} 
are the indices of the ratios of numbers to one another ; being a 
series of numbers in arithmetical progression, corresponding to 
others in geometrical progression. 

-,, C 1 2 3 4 5 indices or Logarithms, 

lftus I 1, 10, 100, 1000, 10000, 10000Q, 

This is the most convenient series of numbers, to which most 
of the modern tables of logarithms are calculated. 

In which it is apparent that if any two indices, or Logarithms, 
be added together, their sum will be the index, or .logarithm, of 
that number, which is equal ta the product of the two terms, in 
the geometrick progression, to which those indices, pr loga- 
rithms belong. 
Thus, the logarithms 2 and 3, being added together,make 5, cor* 
responding to 100000, the product of 100, into 1000, and the 
logarithms 1, and 4, being added together, make 5, the loga-* 
rithm corresponding to 10000C, the product of 10 into 10000, 
Whence it is evident that powers of the same root may be 
multiplied) by -adding their exponents, or logarithms. In like 
manner, if any one index, or logarithm, be subtracted from 
another, the difference will be the logarithm of that number* 
which is equal to the quotient of the two terms, to which those 
logarithms belong. Thus j if from 5, (the logarithm of 100000) 



ARITHMETICS. 45 

be subtracted 2, (the logarithm of 100) the difference 3, will cor- 
respond to 1000, the quotient of 100000 divided by 100. 

Again; if from 5, (the logarithm 100000) be subtracted 3 9 
(the logarithm 1000) the difference is 2, answering to 100, the 
quotient arising from 100060 divided by 1000. Hence it is 
manifest, that. ' ^ 

, A power may be divided by another power of the same root, 
%y subtracting the logarithm of the divisor, from the logarithm 
>f the dividend. 

,J So also if the logarithm of any number be multiplied by the 
index of its power, the product will be equal to the logarithm 
of that power. Thus if 2, (the logarithm 100) be multiplied by 
3, the product will be 6, equal to the logarithm of 1 000000, or 
the 3d power of lot). 

Again, if the logarithm of any number be divided by the in- 
dex of its root, the quotient will be the logarithm of its root. 

Thus, the index, or logarithm of 1000000, is 6, and if this 
number be divided by 3, the quotient will be 2, which is the 
logarithm of 100, or the cube root of 1 OOOpOO. 
In the following series, to wit. 
10* : 10 5 : 10* 10* 10° 10^ x 10-* JO-** 10-* 
t JOQOO 1600 100 10 1 .10 100 , 1000 10000 
T Whose Logarithms are 

4 3 2' 1 — 1 , ' —2. —3 *-4 &c. 

It will be seen, that the logarithms of all the numbers be- 
tween 1 and 10, are greater than 0, but less than 1 ; since by the 
series, it may be seep, that the logarithms of 1 and of 10, are O, 
and 1. 

; Thus the logarithm 
of 2 is 0. 3010300 

of 5 == 0. 6989700 

, of 7= 0. 8450980 

Each number therefore between I and 10, has for its index, 
with a decimal annexed. [ ' 

Eor the same reason, if the given number be 
V between ") the log. f\ and 2 1 + the decimal part 

ISoind^OOO } **» be \ 2 and 3 L e - 2 + the decimal *** 

1000 & 10000 J between t 3 and 4 3 + the decimal part 

Thus the logarithm of the natural number 

of 35 is \ 1. 5440680 

of 175 is 2. 2430380 

of 8795 is 3. 9442358 

Whence' we derive this general truth. ^The index of the 

logarithm, is always 1 less than the number of integral figures 

in the natural' number^ whose logarithm is required ; or the in- 



46 LOGARITHMICK 

dex shows how many figures to the left, the natural number ex* 
tends from the place of units. 

Thus the logarithm of 35, is 1. 5440680. Here the number 
of figures being two> the index or characteristic!*, ef the log- 
arithm is 1. 

The logarithm of 175 is 2 2430380. Here the nuiriber 17* 
consists of three Jigures f the first of which on the left hand is 
second from the place of units ; the index, or characteristick, 
of the logarithm is therefore 2, and the logarithm 8795 is 3. 
9442355. Which extending to three places counted from the 
unite figure, must have 3 for the index of its logarithm. ( 

Integral numbers are said to form a geometrical series, in- 
creasing from unity towards the left; but decimals are supposed 
to form alike series, decreasing from unity towards the right? 
the indices of whose logarithms are negative as has been shown 
in the preceeding examples. 

Whence it follows, that all numbers which consist of the same 
Jtgures, whether integral, or fractional, or mixed, will have the 
decimal parts of their logarithms the £ame y differing only in the 
index, which will bemonr, or les8 y and fioritivey or negaiive r 
according as the first .figure of the number is removed to the 
right, or left, from the place of units. 

Thus the logarithm of 735? is 3. 8668188 ; and the logarithm* 
of ^y or t i v or tjfes &c. part of it, will be as follows. 



Numbers. 


Logarithms. 


7359 


3. 8668181 


Ui 735.9 


2. 8668188 


73.59 


1, 8668188 


7.359 


0. 8668188 


.7359 


— I. 8668188 


.07359 


— 2. 1668188 


,007359 


— 3. 8668188 



Thus it appears that the,aegative index of a logarithm, s^ows 
how far the first significant figure of the natural number, is re- 
moved from the place of units on the right, in the same manner, 
as a positive index shows how far the first figure of the natural 
number, is removed from the place of units op the, left. But 
wl?en the index of the logarithmic negative, it is oftenmore con- 
venient to make it fiositive ; and this is done by adding 10 to 
the negative index. 

Thus, instead of — 1. 8668188") f 9. 8668188 

of — 2. 8668188 IwriteX 8. 8668188 
Of — 3. 8668188J 17.8668188 

Bpcause -*- i + 10 = 9, and ~ 2 + 10 = 8, and ~m 3+ 10 /• 
Although this in truth, makes the index 10 too great ; yet by a lit- 
tle caution, it will pvoduce no ewour in the result ;— observing 



ARITHMETICS At 

always,; that when thesum, or product of the indices so increased, 

exceeds 10, 10 must be rejected. 
Thus the sunx of 
— 2. 8668188 > . _ . C8. 8668188 
_ 3. 8668188^ bCCOm<?S 2 r. 8668188 



» _4. 7336376 6. 7336276 

For— 4 +.10 = 6 



And the product of — 3. 8668188" 
Multiplied by 2 2 



5. 7336376 
For— .5 + 10 == 5 



>or< 



7. 8668189 
2 



CONSTRUCTION OF LOGARITHMS. 

The usual method of computing the logarithms to any of the 
natural numbers, 1. 2. 3. 4. 5. Sec. is, I believe, as follows. 

RULE.f 

1. Take any two numbers whose difference is unity, or 1, 
and let the logarithm to the lesser number be known. 

2. Divide the constant decimal, 868588964, &c. (or, 2-f — 2. 
3025, Sec.) by the sum of the two numbers, and reserve the" quo- 
tient ; divide the several quotients by the square of the sum of 
the two numbers, and reserve the quotient ; divide this last 
quotient also, by the square of their sum, and again reserve the 
quotient ; and thus proceed, continually dividing the last quo- 
tient, by the square of the sum Of the two numbers, aslong as divi» 
sion can be made. 

3. Then write these quotients in their order, under one anoth- 
er, the 1st uppermost ; and divide them respectively by the 
prime, or odd numbers, 1. 3. 5. 7. 9. 11. 13. 8c c. as long as di- 



•The decimal parts of these logarithms are added as in sim- 
ple numbers ; but when you come to the left hand figure of 
each deeimal, there is -h 1 to carry to — 3, which equals — 2, 
and this added to the — 2 above it,* gives — - 4 for the sum of 
the indices. 

t Yet there are many other ingenious methods of finding the 
logarithms of numbers, (see Introduction to Dr. Huttons Tables, 
and Baron Maseres Scriptores, Logarithmici, also, Kiel, on 
Logarithms, Briggs Logarithms, Gardners, Taylors, Callets* 
and Sherwins, Mathematical Tables. 



i 



48 



LQGARITHMICX 



vision can be made, that is, divide the first reserved quotient by 
1, the second by 3, the third by 5, the fourth by 7, and so on. 

4. Add all these last quotients together, and the sum will be 
the logarithm Of the greater number divided by the less ; there- 
fore to tl)is logarithm, add also the logarithm of the lesser num* 
ber, and their sum will be the logarithm, to the greater, or pro- 
posed number. 

EXAMPLES. 

Jlx ample 1st. Let it be required to compute the logarithm 
of the number 2. 

Here the given, or greater number is 2, and the next less 
number is 1, (whose Logarithm, in every System, is always 0) 
*lso the sum of 2 and 1 is 3. and its square 9 ; as follows. 



3) 
9) 
9) 
9) 
«) 
9) 

a 

9) 



868588964 

289529654 

32169962 

3574440 

% 397160 

44129 

4903 

545 

61 



1) 

■o 

•) 
H) 
13) 
15) 



289529654( 

32169962( 

3574440( 

397160( 

44129( 

4903( 

545( 

61( 



289529654 

10723321 

714888 

56737 

4903 

446 

42 

4 



Log. of 2—1. 
Add. log. of 1 



.301029995 
.000000000 



True Log. of 2. .301029995 

Example 2rf. 
Let it be required to compute the logarithm of 3 
Here the given number is 3, and the next less is 2, whose 
logarithm by the first example is .301029995, and the sum also, 
of the 2 numbers 3 + 2=5 the square of which is 25, then the 



operation is as follows. 



*) 
25) 

25) 

25) 

25) 

25) 

25) 



868588963 

173717793 

6948712 

277948 

11118 

445 

18 





*> 

5) 

7) 

9) 

H) 



173717793( 

69487 12( 

277948( 

11118( 

445( 

18( 



173717793 

2316237 

55590 

1528 

50 

o 



Log. of 3 4- 2 
Add Log. qf 2' 



.176091260 
301029995 



True log. of 3 required. 477121255* 

• See « Mr. Hutton's practical rule for the constvtfctipa-.of 
logarithms.'' 



Arithmetic*. 



49 



Iflien, because the sum of the Logarithms of numbers gives 
the logarithm of their product, and the difference of the Toga- 
arithmsgives the logarithm of the quotient of the numbers ; from 
the above two logarithms, and the logarithm of 10, which is 1, 
we may raise a great many logarithms, as in the following ex- 
amples. 

Example 3d. v Example 4th. 



Because 2X2 = 4, therefore 
To Logarithm of 2 .301029995 [ 
Add Log. of 2 .30 1029995 ; ; 



*fhesumisL6gof4 .602059990 \\ 



Example 5th. 
because 2X4= 8, therefore 
To logarithm of 2 .301029995 
Add log. of 4 .602059990 



Because 2X3= 6, therefore 

To logarithm of 2 .301029995 

Add log. of 3 .477121255 



Che sum is Log. of ,6 .778 1 5 1 250 



Example 6th. 
Because 10-^-2=5, therefore 
From log. of 10 1.000000000 
' r uke log. of 2 .30 1 029995 



Give* the log of 8, 903089985 : Remains log. of 5 .69897000* 



Example 7th. 
Because 5 X 8 = 40, thereifore 
To logarithm of 8 .903089985 ■ 
Add log, of 5 .698970005 



Example 8th. 
Because 8 X 40 = 320 therefore 
To Logarithm 40 1 60205999* 
Add log. of 8 .90308998ft 



Logarithm of 320 2 50514997* 



Logarithm of 40 1.6O205999O 

And thus computing by this general Rule, the Logarithms t* 
"the prime numbers, 2. 3. 7. 11. 13. 17. 19. 23. 29. 31. 37. 41. 
43. 47. 8cc. and then by using composition and division, we may 
easily find as many logarithms as we please^ or examine any log-* 
arithm in the table* 



ANOTHER METHOD op COMPUTING LOGARITHMS' 



The construction of Logarithm according to the proceeding 
Irules, given by the repeated extraction of Roots, is tedious \ the 
simplest method yet known is the following. 

* To make a Table of Logarithms* 
\. Write for the logarithm of 1, a cypher for the index, and 
as many cyphers for the decimal part of the logarithm as you 

* See "Mr. Briggs differential method of constructing logf* 
jMJthms." 



*» LOGARITHMIC* 

would wish the logarithms to be extended : for tire logarithm 
of 10, write an unit, with the same number of cyphers ; for the 
logarithm of 100, put 2, with as many cyphers ; for 100(5, put 3 
for the index, with as many cyphers; for 10000, put 4 for the 
index, &c. 

2. Find the difference between some two logarithms above 
1000, or rather 10000, that differ by unity ; multiply the two 
numbers together, and that product by the constant decimal 
43429448190325183896 &c. 

3. Divide the last product by the Arithmetical mean between 
the two numbers, and the quotient will be the logarithm of the 
difference of the two numbers. 

Thus ; Let it be required to find the difference between the 
logarithm of 10000, and 10001. The product of these two 
numbers is 100010000, which multiplied by 4343, &c. gives 
43434343, which divided by 10000.5, the Arithmetical mean 
between the two numbers, gives 4343. Now if to the logarithm of 
1000 which is 4. 0000000,43 43 be added, we shall have 4.0000434, 
the true logarithm of 1000 to 7 places. 

Having thus found the difference of the logarithms of any 
two numbers differing by unity, or 1, and consequently, some of 
the logarithms, by dividing the difference found by the Arith- 
metical mean of any two numbers differing by 1, we shall have 
the difference of their logarithms. 

Thus ; to find the difference between the logarithm of 274 t 
ifcnd 275, divide 4343, the difference of the logarithm of 10000, 
and 10001, by 274.5 the quotient will be 1582, the difference 
required, which added to 2. 4377506 the logarithm of 274, gives 
2. 4393327 for the true logarithm of 275. 

5. Having by this Means found a few of the* prime logarithms, 
the rest are made by Addition and Subtraction ; and having^ 
made the canon upward, above 1000 to 10000, by consequence 
it is made for all infer io» numbers. 

Directions for taking Logarithms nnd their numbers 
from the Table. 

To find the Logarithm of any number consisting of 4 figures* 

Look for the number whose logarithm is required in the col- 
umn of numbers, and against this number^ its logarithm will be 
found. , ^ 

Thus ; the logarithm of 1234, is 3,0913151, so that* any num- 
ber under 10000, may be easily found by inspection. 

But if the number is greater than 10,000, but less than 
10,000,000 

*Cut off four figures on the left of the given number, and seek 
the logarithm in the table ; add as many units to the index, as 



AMTHMETICIU 61 

there afce figures regaining on the right : subtract the logar- 
ithm found, from the next following it in the table :— -then sis the 
difference of numbers in the Canon, is to the tabular distance of 
the logarithms answering to them, so are the remaining figures 
of the given number to the logarithmick difference : which if 
it be added to the logarithm before found, th* sum will be the 
logarithm required. Thus, let the logarithm of the number 
92375 be required. Cut off the four figures 9237, and to the 
index of die logarithm corresponding to them, Add one unit, be- 
cause one figure is cut off on the right. 

Then from the logarithm of the next greater, 

number. 9238, = 3.965578a 

Subtract the logarithm of the 

required number, 9237 = 39655309 

* 10 471 

♦Then as 10 : 471 t : 5 : 235 

Now to the log. of 4.9655309 

Add 235 the difference found. 



And the sum is the log. 4.9655544 required. 

Or more briefly ; find the logarithm of the first four figuru* 
as before ; then multiply the common difference which stands 
against it, by the remaining figures of the given number, from 
the product cut off as many figures at the right hand, as you 
multiplied by, and add the remainder to the logarithm before 
found, fitting it with a proper index. 

thus, 471 X 5 = 2355, cut off 5, and add 235. 

To find, the Logarithm of a dicimal fraction. 
The logarithm of a dicimal, is the same as that of a whole 
number excepting the index. 

Take out then, the logarithm of a whole number consisting 
ef the same figures, observing to make the negative index 
equal to the distancetof the first significant figure ofthe fraction,. 
from the place of units. 

The log. of 0.07643 is — 2. 8832639 f or 8. 8832639 
of 0.00259 —3. 4132998*1 or 7. 4132998 

of 0.0006278 —4. 7978213 t or 6. 7978213 

To find the logarithm of a mixed dicimal number. 
Find the logarithm, in the same manner as if all the figures 



\ 



*If one figure is cut off, say as 10 is to the diff. of Log. if 
two figures are cuteff, as 100 is to the diff. if three, as 1000 ? 
Sec. 



*3 fct>GARlTHMrc£ 

were integers ; and then prefix the index which belongs to th> 
, Integral part. 

Thus, the logarithm of 39.68 is 1. 5985717 
Here the index is one, becouse 1 is the index of the logar 
rithm of every number greater than K>, and less than 100. 

lo find the logarithm of a vulgar fraction* 
Subtract the logarithm of the denomenator from that of the, 
numerator. The difference will be the logarithm of the frac- 
tion. • 

To find the logarithm of §£ 

Logarithm of 37 1. 56820 ip 
logarithm of 94 1. 9731279 

Diff. Log. of ££ — 1. 595073$ 
where the index 1 is negative. 

To find in the Table the natural number to any logarithm. 
This is to be done by the reverse method to the former, viz* 
fcy searching for the proposed logarithm among those in the 
Table, and taking out the corresponding number by inspection, 
Id which the proper number of integers is to be pointed off, viz. 
I more than the units of the affirmative index. 

To find the number corresponding to a logarithm greater than 
any in the Table. First, from the given logarithm subtract the 
logarithm of 10, of 100, or 1000, of 10,000, till you have a loga- 
rithm that will come within the compass of the table, find the 
number correspondinjg to this and multiply it by 10, or 100, op 
* 1000, or 10,000, the product is the number required. Suppose 

/ for instance the number corresponding to the logarithm, 

7. 7589875 be required * Y subtract the logarithm of the number 
-* 10.000 which, is 4.0000000 from 7.7589875, there remains 

3.7589875, the number corresponding to which is 5741, thia 
multiplied by 10,000 gives the number answering to the given 
logarithm. 



METHOD OF CALCULATING BY LOGARITHM^ 

MULTIPLICATION, 

RULE. 

Take out the logarithms of the factors from the tabled 
add them together and their sum will be the logarithm of the 
product required. Then by means of the Table take out the 
Natural number answering to the sum for the product sought* 



'9LRITHMETICI5 



& 



EXAMPLES. 



?. Multiply 45 by 27 

Numbers. Logarithms. 
45 1. 6632125 

27 1. 4313638 

product. 1215 3.0845763 Ans. 



<S. Multiply. 

lumbers. Logarithms. 
23.14 1. 3643634 

75.99 1 3807564 



2. Multiply 709 by 13 
Numbers. Logarithms. 
709 2. 8506462 

13 1. 1139434 



9217 



3. 964S896 



Ans. 1758.4086 



3. 2451198 



.Multiply 3.7 by 3. 7 
Numbers. Logarithms. 
3.7 0- 5682017 

3.7 0. 5682017 



Ans. 13.69 



1. 1364034 



5. Multiply 3.586, 2.1046, 0.8372, 0.0294, all together. 
Numbers. Logarithms. 

3.586 0. 5546103 

2.1046 0. 323 1 696 Here the 2 to be carried can- 

0.8372 —,1. 9228292 eels the — . 2 and there re« 

0.0294 — 2. 4683473 mains the —1 to be set down. 



0.1857618 *— 1. 2689564. 



0. What cost 87 pounds of grean tea; at g 2 12 cts.pr. Itt? 
Numbers. Logarithms 

212 0. 3263359 

87 1. 9395193 



Ans. g 184. 44 2. 2658552 

7. What cost 160 bushels oats, at 50cts. pr. bushel ? Ans. g 89- 

18. What cost 250 bushels of wheat, at g 1 60 pr. bushel ? 

Ans. g400 
*'. What cost 1260 lb. rice, at 5 cts. pr. lb. ? Ans. g 65' 



DIVISION BY LOGARITHMS, 

RULE. 
From the logarithm of the dividend, subtract the logarithm of 
the divisor, and the number answering to the remainder will be 
the quotient required, 

* Note. In every operation what is carried from the dicimai 
part of the logarithm, to its index, is affirmative, and is therefore 
V> be added to the index when \t is affirmative^ but subtracted 
■wheji it is negative, 



s* 



ttOGARITHMICE 



EXAMPLES. 



li Divide 15811 by 163 
Numbers. Logarithms. 
Dividend 15811 , 4. 1989593 
Divisor 163 2. 2121876 



2. Divide 163 by B. IS 
Numbers. Logarithm. 
Dividend 163 2. 212187& 

Divisor 8.18 0. 912753$ 



Ans. 



97 I. 9867*717 | quotient 19.926 1. 299434$ 



3. Divide 100000 by 100 

Number. Logarithms. 
Dividend 100000 > 5. 0000000 
Divisor 100 2. 0000000 



Quotient 1000 3. 0000000 



4. Divide 1000000 by l$0Q 

Number. Logarithms. 
Divid. 1000000 6 000000* 
Divisor 1000 3. 0000000 



quotient 1000. 3. OOOOOOfr 



5. A ship took a prize worth 8 3960, it is required to divide it 
equally among the sailors, who are 264 in number. 

What did each man share it the prize ? Ans. 8 IS 

If 125 lb. of sugar cost 2 26. 25cts. what is that pr. lb.? 

Ans. 21 cts. 



PROPORTION BY LOOARITHMSL 



RULE. 

If the proportion be direct, add the logarithms of the second 
ftnd third terms, and from the sum subtract the logarithm of the* 
first term. The remainder will be the logarithm. of the term 
required* 

If the proportion be inverse, add the logarithms of the first 
And second terms, and from the sum subtract the logarithm of 
the third. The remainder will be the logarithm to the required 
term. 

EXAMPLES. 

Find a fourth proportional to 7964, 578, and 27960. 
Numbers. Logarithms. 

Second term 378 2.5774918 

Third term 27960. 4.4465372 



First term 7964 

Fourth term f 327 



7.0240290 
3.9011313 

$: 1228977 



ARITHMETIC** S3 

Find a fourth proportional to 768, 3frl, and 9780,. 
Numbers. Logarithms. 

Second term 381 2.5809250 

Third term 978* 3.9903389 



6,5712639 
First term 768- 2.8853612 



Fourth term 4852 3.6859027 



♦ARITHMETICAL COMPLEMENT. 

The difference between a given number, and 10, or 100, or 
1©00, Sec. is called the Arithmetical Complement of that 
number. 

To obtain the Arithmetic At Complement of a number, 
subtract the right hand significant figure from 10, and each of 
the other figures from 9. 

N. B. In taking the Arithmetical Complement of a logarithm, 
if the index is negative, it must be added to 9 ; for adding a nega- 
tive quantity, is the same as subtracting a positive one. The 
difference between — 4 and + 8, is not 4, but 12. 

THE ARITHMETICAL COMPLEMENT. 

Of 5. 2473621 is 4. 7526379 ' 

of 1.9864362 is 8.0135638 

of ©.6452310 is 9. 3547689 

of-2. 7064923 isl 1.2935076 

In the foihiDing firoft^rtion^ the calculation is made in botlp 

ways, 
1. If the profit on g 2625 employed in trade, is g 525, what is 
the profit on g 7875 ? 



* Note. The principal use of the Arithmetical Complement, 
is, in working proportions by Logarithms ; for by this they may 
be performed by merely adding' together the several terms of the 
proportion. 



*tf tOGARITHMICE; 



By the common method. 
Second term 525 2.7,201593 
third term 7875 3.8962506 



6.6164099 
First term 3625 S.4191293 



3y the Arith'l.Compilemerik ^ 
Second term 525 2.7201 59 » 
Third term 7875 3.8962506 

First term, a. c. 6.580870f 



Fourth term g 1575 .3.1972806 



Fourth term 1575 3.197280$ 



2. If g 567, gain g 81, what will be the gain on g 1701 ? 
As 567 stock a,c. 7.2464169 

Is to 81 profit 1.9084850 

So is 1701 stock 3.2307043 



To 243 profit 2.3856062 

5. If the interest on g 450 for one year, is g 2?, what will be* 
the interest on g 1150 for the same time. Ans. g 69 ? 

4. When a pipe of wine costs g 252, what is the value of 174 
gallons ? Ans. g 35 ? 

5. Bought 721 yards broad cloth, at the rate of g 65 for every* 
13 yards, what did the whole come to ? Ans. g 3605 ? 



VULGAR FRACTIONS. 

A vulgar fraction is any assignable part of a unit or integer, 
expressed by two numbers, placed one above the other' with a 
line drawn between them, as J one fourth, «£■ two thirds, Sec. 

The number above the line is called the numerator, and that 
below the line, the denominator. 

A fraction is said to be in its lowest terms, when it is ex- 
pressed by the least numbers possible, as £, when reduced to its- 
lowest terms will be \ ; and T 9 T is equal to \ Sec. 

CASE I. 

To reduce fractions to their lowest term*. 

rule. , 

Divide both the numerator and denominator, by any number 
Which will divide them without a remainder, and the quotients 
again in the same manner, till it appears that there is ho numbetf 
greater than 1,' which will divide them again,. 

EXAMPLES. 

1. Reduce -JtI to its lowest terms. Ans. \ 

3. Reduce '-f^ to its lowest terms. Ans. -J* 

3. Reduce f? to its lowest terms. Ans.. %* 



ARITHMETIC** 5/ 

4. Reduce |ff to its lowest terms. Ana. { 

5» Reduce -f J$£ to its lowest terms. Ans. J 

Abbreviate $ffflfi as much as possible. Ans. 5SV1SV 

CASE II. 
To reduce the value of a fraction to the known parts of an 
integer. 

RULE. 
Multiply the numerator by the common parts of the integer 
and divide by the denominator. 

EXAMPLES. 

1. What is the value of f of a pound sterling ? 
2 
20 shillings in a pound. 

Denominator 3)40(1 3s. 4d. Ans. 



10 
9 

_ * 

1 
12 pence in a shilling. 

3)12(4<k 
12 

2. What is the value of f pound sterling ? 

Ans. £0 15s. O 

3. Reduce { of an hundred weight to Us proper quantity. 

Ans. 3qrs. 31b. loz. 12$dr, 

4. Reduce f of lb. troy to its proper quantity. 

Ans. 7oz. 4pwt. 

5. Reduce $ of a mile to its proper quantity. 

Ans. 6 fur. 16 po. 
6% Reduce § of a month to its proper quantity. 

Ans. 2 wks. 2d. 19 h. 12 m. 
CASE III. 
To reduce a fraction of one denomination to that of another, 
but greater, retaining the same value. 

RULE. 
Reduce the given quantity to the lowest term mentioned, for 
a numerator ; tfcen reduce the integral part to the same term, 
for a denominator 5 which will form tfcte fraction required. 



8 



58 LOGARITHMICK 

EXAMPLES. 

1. Reduce 15s. 8d. 2qrs. to the fraction of a pound. 
20 Integral part. 15.82 given sum. 

12 12 

240 188 

4 4 



»60 ueuominator. 754 JNum. jfj = f ±%£ . 

2. What part of a pound sterling is 13s. 4d. ? Ans. § 

3. What part of a hundred weight is 3 qrs. 14lb. ? Ans. f 

4. What part of a yard is 2 qrs. 1 nail ? Ans. T * F 

5. What part of a common year is 3 weeks, and 4 days ? 

Ans. T % 

6. What part of a mile is 6 fur. 26 po. 3 yds. 2ft.? 
fur. fio. yd. ft. feet. 

6 26 3 2 = 4400 Num. 

a mile = 5280 Denom. fffj = £ 

7. What part of a hhd. of wine is 54 gallons ? Ans. 4 

8. What part of a day is 16 h. 36 min. 55 T 5 j sec. Ans. T 9 ^ 

9. What part of a shilling is 4Jd. ? Ans. | 

10. What part of an acre is 3 roods, and 20 rods ? Ans. £ 

1 1. What part of a pound troy is 10 oz. 11 pw*. 16 grs ? 

Ans. if J 

12. What part of a cord is 1 16-A- feet ? Ans. j-j 

— -/> — — 
DECIMAL FRACTIONS. 

A Decimal Fraction is that whose denominator is an unit, 
with a cyper, or cyphers annexed to it, thus T % T fo T7 5 ^ &c. 
&c. 

As the integer is always divided either into 10, 100> 1000 &c. 
equal parts ; consequently the denominator will always be 
either 10, 100, 1000, 10,000, &c. which being implied, need not 
be expressed ; for the true value of a Decimal Fraction is prop- 
erly expressed by writing the numerator, only with a point be- 
fore it on the left. 

f * 1 f 5 

Thus, instead of < T 7 / 7 > write «< .75 

I iW* J ' I -837 &c. 

But if the denominator has not so many places as the denomi- 
nator has cyphers, prefix so many cyphers on the left as will 
make up the deficiency. * . 



ARITHMET1CK. 59 



Thus, for -{ u \ r > write 4 .C 
L xoJor J l < 



.05 
.006 

.0007 &c. 
Decimals are reckoned from the left hand towards the right, 
and the value of each figure is determined by its distance from 
the place of units , if it be in the first place after unit,s (or sep- 
erating point) it signifies tenth's ; if m the second, hundreth's 
&c.dec reasing towards the right in a tend fold proportion, as in 
the following 

TABLE. 

Crt (A 

*•» «-> 

2 as, 

G c: 2 A.J3 CCA 

(0 3 3 CJ^ ft*? fi 3 3 w 

C O O rt £ £ «* O © C 

2HH0CC3 cgonHn: 

SotfHKH^ HKHXoS 

,765 4321 234567 

* ^ 1 V ^-— J 

Whole Numbers. Decimals. 

Cyphers placed at the right hand of a decimal fraction do not 
alter its value, since every significant figure continues to pos- 
sess the same place ; so, 5, 50, 500, 5000, are all of the same 
value, and each equal to / T or t 

But cyphers placed at the left hand of decimals, decrease 
their value in a ten fold proportion, by removing them farther 
from the decimal point. Thus, .5 .05 .005 .0005 Sec. are 
5 tenth, 5 hundredth, 5 thousandth paats, &c. respectively 



ADDITION OF DECIMALS. 

RULE. 

1. Place the numbers, whether mixed, or pure decimals,, 
under each other, according to their local value. 

2. Find their sum as in whole numbers, and point off so many, 
places for decimals, as are equal to the greatest number of de- 
cimal parts in any of the given numbers. 

Note. The point prefixed is called the Seperatrix, from, 
its separating the Integral from the Decimal part. 



40 liOQARITHMICK 



EXAMPLES. 

1. What is the sum of 276, 39.213, 72014.9, 417, 5032 and 
2214.298? 

/ , 276. 

39.213 
i 20 14.9 
417. 
5032. 
2214.298 



AhS. 79993.411 



Etence we itiay observe that the Denominations of Federal 
Monet, as determined by an Act of Congress Aug. 8, 1786, are 
in a decimal ratio, and subject to one, and the same law of nota- 
tion, and consequently of operation. 

For since a dollar is the integer or unit , and a dime being 
the tenth, and a cent the hundredth, and a mill the thousandth 
part of a dollar, it is evident that any number of dollars, dimes, 
cents, and mills, is simply the expression of dollars, and deci- 
mal parts oi a dollar : Thus, 15 dolls. 8 dimes, 3 cents, 5 mills 
is expressed in decimals. 15,835, or IS.-fJfo 

2. What is the sum of the following expressions of money, 
viz. R7530, g 16.201, g 3.0142, g 957.13, g 6.72819, g .03014 ? 

Ans. g 8513.10353 

3. Required the sum of .01 4, .9816, .32, .15914, .72913, and 
.0047, Ans. 2.20857 

4. What is the sum total of 27.148, 918.73, 14016, 294304, 
.7138, and 221.7 ? Ans. 309488.2918 

5. Required the sum of 512.984, 21.3918, 2700.42, 3.1£& 
57.2, and 581.06, Ans. 3646.20888 



SUBTRACTION OF DECIMALS. 

RULE. 

1 . Set the less number under the greater in the same manner 
as in addition. 

2. Then subtract as in whole numbers, and place the decimal 
point in the remainder directly under the other points. 



ARITHMETIC'S. 



61 



Dollars. 
From 21.481 
Take 4.90142 



EXAMPLES 



Rem 

3. From 

4. From 

5. From 



Feet 
From 125.64 
Take 95.5875$ 



Rem. 30.05244 
Ans. 194.7925 



16.57958 
270.2 subtract 75.4075 
2.73 subtract 1.9185 

.9173 subtract .2138 

6. From 407, subtract 91.713 

7. From 800.135 subtract 16.37 

8. What number added to 9.999999 will make 10? 



Ans. 
Ans. 
Ans. 
Ans. 



o.ans 

.7035 
315.287 
783.76$ 



Ans. One millionth part of an unit? 



MULTIPLICATION OF DECIMALS. 



RULE. 

1. Set down the factors under each other, and multiply them 
as in whole numbers. 

2. And from the product) on the right, point as many figures 
for decimals, as there are decimal places in both the factors. 
But if there be not so many figures in the product as there 
ought to be decimals, prefix the proper number of cyphers to* 
supply th£ defect. 

EXAMPLES. 



1. Multiply 91.78 
by .381 



Multiply 520.3 
by .417 



product 34.968 1 8 product 2 1 6.965 1 

3. Multiply .217 by .0431 Ans. .0093527 

4. Multiply 5 1 .6 by 2 1 Ans. 1083.6 

5. Multiply .051 by .0091 Ans. 0004641 

6. What will 6.21 yards of cloth amount to, at 2 dollars 32 
cents 5 mills pr. yard ? Ans. 2.325 X 6.21 =836.9954 

7. What cost 27.13 lb. of green tea, at gl 12-J pr. lb. ? 

Ans. 830.52125 

8. What cost 53% lb. sugar^at 14-J cents pr. lb. ? 

Ans. 87.79375 

9. What will 12.125 acres amount to, at 865.25 pr. acre ? 

Ans. 8791.15625 

10. What is the value of .7584 ounces oi gold, at %\7.777 pr 
oz. ? Ans. 813,482076'8 



62 LOGARITHMICK 

To multiply by 10, 100, 1000, &c. remove the separatrix, in 
the multiplicand, so many places to the right, as the multiplier 
has cyphers. 

examples. 

'7.853 
78.53 



The product of .7853< 



Into 10 = 7.8530") 
" 100= 78.5300* n 
" 100a = 785.3000 f UV 
J' 10000 =.7853.0000J 



785.3 
7853 



DIVISION OF DECIMALS. 

RULE. 
Divide as in whole numbers ; and to know how many deci- 
mals to point off in the quotient, observe the following rule. 

1. There must be as many decimals in the dividend, as in 
both the divisor and quotient ; therefore point off for decimals 
in the quotient so many figures, as the decimal places in the di- 
vidend exceed those in the divisor. 

2. If the figures in the quotient are not so many as the rule 
requires, supply the defect by prefixing cyphers. 

3. But if the decimal places in the divisor exceed those in 
dividend, add cyphers as decimals to the dividend, till the num- 
ber of decimals in the dividend be equal to those in the divisor, 
and the quotient will be integers till ail these decimals are used. 
And, in case of a remainder, after all the figures of thj&dividend 
are used, and more figures are wanted in the quotient, annex 
cyphers to the remainder, to continue the division to any degree 
of exactness. • 

EXAMPLES. 

1. Divide 5424.6056 by 43.6 

43.6)3424-6056(78.546 
3052 




2580 
8180 



2616 
2616 

remains. 



ARITHMETICS, /a 

2. Divide 3877875 by.675 Ans. 5745000 

3. Divide 7.25406 by 957 Ans. .00758 

4. Divide 56 cts. by 1 doll. 12 cts. Ans .5 

5. If 6.2 1 yards of cloth cost g 14.43825, what cost one yard ? 

Ans. 22.325 

6. What is the value of 1 lb. green tea, when 27.131b. cost 
230.52125 Ans. 8 .12$ 

7. If an ounce of gold be worth g 17.777, what is the value of 
.7584 of an ounce at the same rate ? Ans. £13.4820768 

Note. When decimals, or whole numbers, are to be divided 
by 10, 100,1000, 8cc. remove the separating point in the divi- 
dend, so many places towards the left, as there are cyphers in 
(he divisor. 

EXAMPLES. 

10 the quotient is 785.3 
100 -' " 78.53 



by V 



7853 divided „, ^ 10()0 „ „ 7 g53 

10000 " " .7833 



j 



REDUCTION OF DECIMALS, 

CASE I 

To reduce a Vulgar Fraction to its equivalent Decimal. 
RULE. 
Annex i^phers to the numerator, and divide by the denomi- 
nator, the quotient will be the decimal required. 

Note. So many places must be pointed off in the quotient, as 
there were cyphers annexed to the numerator ; but if there be 
not so many places of figures in the quotient, supply the defect 
by prefixing cyphers on the left of the said quotient.. 

EXAMPLES. 

i*. Reduce <§• to its equivalent decimal. 
8)1.000 

.125 Arcs. 

2. Reduce £ to a decimal. Ans. .25 

3. Reduce \ to a decimal. , Ans. .5 

4. Reduce \ to a decimal. Ans. .2 

5. Rednce \\ to a decimal. Ans. .85 

6. What decimal is equivalent to ^V? Ans. .025 
.7. Bring ^ T to a decimal. Ans. .09375 
&. Find the decimal expression of ttit' Ans * ,co8 



i 



*4 LOGARITHMICK 

CASE II. 

To reduce number* of different denominations to their equiy* 
alent decimal vplue*. 

RULE. 

Turing the given denominations to a vulgar fraction (by Case 
IIJ. page 57) and reduce said fraction to its equivalent decimal 
value. 

OR 

Rule 2. Write the several denominations above each other in 
their order, placing the highest denomination at the bottom ; 
tfonj beginning at the top, divide each denomination by its val- 
ue in the next superiour denomination ; the last quotient will 
be the decimal required. 

EXAMPLES. 

1? Reduce 15s. 9d. 3qrs. to the decimal of a pound. 
15 
12 

189 
4 

2Q X 12 X 4 = S60)759.000000(.790625 Answer. 
6720 



8700 
8&Q 






6000 


.« 


5760 




_ 


By RULE 2 


2400 


4 


3, 


1920 


12 


9.75 


»»— 


20 


15.8125 



4800 



4800 Decimal .790625 required. 

2. Reduce 12s. 6d. 3qrs. to the decimal of a pound. 

Ans. .628125 

3. Reduce 9s. to the decimal of a pound. Ans. .45 

4. Reduce 19 s. 5d 2 qrs. to the decimal of a pound. 

Ans, .9727916 

5. Reduce 3s. 9d. to the decimal of a dollar. Ans. .625 

6. Reduce 7 oz. 19 pwt. to the decimal of alb. Troy. 

Ans. .6§25 

7. Reduce 3 qrs. 2 1 lb. to the decimal of an c wt. 

Ans. .93?[5 



ARITHMETIC*. 



SIMPLE INTEREST. 

Ikterest is the premium allowed for the loan of money, relr 
alive to which there are four particulars. 

First, the Principal, or sum at interest. 

Second, the Rats per cent, or interest of /*100, or dollars 
for one year. 

Fourth, the amount which is the sum of principal apd inter- 
est, added together. 

Interest is either Simple, dr Compound. 

Simple Interest is that which arises from the principal 
only. 

RULE. 

1 . Multiply the principal by the rate, and divide the product 
by 100. The quotient is the answer for one year. 

2. Multiply the interest for one year by the given number of 
years, and the product is the answer for that time. 

$. If there be parts of a year, as months, or days, work for the 
months by the aliquot parts of a year, and for the days by simple 
proportion, 

Note. Solutions in Simple Interest exhibit the principle* 
that the interest of £ 100, or £100, for 1 year, 2 years, 3 years, 
Sec. correspond to a series,©/ numbets in arithmetical propor- 
tion ; from whence will naturally arise the following Theorem* 
that 

If two ranks of numbers have the same ratio between every 
pair of correspondents, then the numbers themselves, their cor- 
respondent sums, and correspondent differences, will have th« 
game common ratio. 



Tbits 



{ 







Principle. Int. 


1 : 6 




100 t fc 


2 : 12 




200 I 1* 


3 : 19 


or 


300 : 1* 


4 : 24 




400 : 24 



10 : 60 1000 : 60 

In the first pair of ranks, the ratio between any two corre*> 
pondents, is 6 ; therefore, taking any number from the first 
rank,— suppose 2 ; then 2 : 12 : : 10 : 60; and 2 t 12 v z 
£ : 48 ; or 2 : 12 : : 3 : 18. So also in the 2d. pair of, 
rank's ;— 300 : 18 : : 400 : 24 \ and 300 :- 18 : : 1*9* : 6# > 
or 300,: 18 : : 7(fe : 42 &cv 
& 



v £# ' JXXJARfTHMICK 

EXAMPLES. 

What is the interest pf £420 for 1 year, *\t £7 pr. cent, pir. 
annum. Ans. r29 8*. 

(1) 

420 What is the interest of fltiS 16*. for 
7 * 1 year, at 7 pr. cent ? Ans. £52 4*. 1 JoT. 

.r- ( 2 ) 

29 | 40 745 16 

20 7 



« 



8 | 00 . ^ 52 J 20 12 

Ans. £29 8*. 20 

4 J 12 
12 

1 J 44 

4 

1 | 76 

3. What is the interest of £800, for 1 year, at 7pr. cent, w: 
•nnum ? Ans. £56 

< 4. Wiiat is the interest of £ 7 6, for 1 year, at 5 pr. cent ? 

Ans. ^3 16*. 
• 5. What is the interest of £2 11 5* for 1 year, at 7 pr. cent? 

Ans. £14 15*. 9rf. 

6. What is the interest of £472 1*. for 1 year, at 7 pr. cent, 
pr. annum f^ Ans. £33 0*. lOJrf. 

7. What is the interest of £270 10*. 6d. for 1 year, at 5 pr. 
•ent, pr. annum? Ans. £13 10*, 6£rf. 

8. What is the interest of g 542, for 1 year, at 7 pr. cent, pr. 
annum ? * Ans. g 37 »94cts. 

9. What is the interest of g 800, for otie year, at 6 pr. cent ? 

, Ans. &48 00 

10. What is the interest of g 875 35ceMs, for one year, at 
6 pr. cent I Ans. g 52 52 

11. What is the amount of a bond for g 387 50 cents, for one 
year, at 6 pr. cent ? Ans. g 410 75cts. 

Note 1. When the principal consists of dollars, multiply ky 
the rate pr. cent ; the product will be the interest for 1 year, in 
cents. 

Note 2. Wfeen the amount is required, add the principal to 
the interest. 



ARITHMETICS. %f 

CASE It. 

1. If the interest required be for, years, months, end days, take 
| the number of months, and set it under the place of tens, take 
i part of the number of days and put it under the place of units 
for a multiplier. 

2. For the odd days, (if any) see what proportion they bear to 
the week, and divide the prinoipal by this proportion, and then 
proceed to multiply as in whole numbers ; the product will bo 
the interest for the whole time, in dollars, cents, and mills. 

EXAMPLES. 

Required thfe interest of g 10 44cts. for 3 years 5 months and 
10 days, at 6 pr. cent, pr. annum. 

. y.m. tf. 

f *= -ffil 10 44 3 5 10 

, 2 06 12 

i | 41 

20+ 1 as 30 days. 
10 

6 I 40 [ 6 
2,15,7,60 Ans. 2 dol. 1$ cts 36 
■ „.',- 7 mills, T^tn. — 

4 odd days = ^ of 
. ./ ' • week. 

2. What will 780 dols.. amount to, at 6 pr, cent, in 5 years 7 
months and 12 days? Ans. 8 975 99cts. 

3. What is the interest of g 824 15 cts. for 22 weeks, at 7 pr. 
fcent ? Ans. g 24 40 cts. 7 m. 

4. What is the interest of g 438 24 for 4 years 9 months and 
14 days, at 7 pr. cent ? Ans. g 146 90cts. 7m. 

CASE III. 
When there is a fraction as %\ £, &c. in the rate pr. cent. 

RULE. 
Multiply the principal by the rate pr. tent, to the product add 
£,£, &c. of said principal, and divide by 100 for the interest re- 
guirerf. 



3 


48 


3 


48 


62 
K>88 


64 



<# BOGARITHMICK 

EXAMPLES. 

1. WhMis the interest of 8 428 for ©fid y*ar> *t*| pr. cent, 
fr. annum? 

51*28 
6* 



2568 
t | 214 

ior 



for half.' 

for one fourth. 



g 28,8$ cts. Ans. 
*. What is the interest of 216/* 5*. for one year, at 5f pr.* 
t«nt ? , Ans* 11/, 17*. H<i 

3. What is the interest of S 300 for one year, at 6 J pr. cent* 
pr. annum ? Ans. g 18 75 cts. 

CASE IV. 
To find the interest of any sum of money, for any number of 
fears and parts of a year. 

1. Find the interest for I year, and multiply this by the giren 
number of years. 

2. If there be months and days, work for the months, by the 
aliquot parts of a year, and for the days, by simple proportion. 

EXAMPLES. 

1. What is the interest of 64 dols. 58 cts. for 3 years 5 month? 
%nd 10 days, at 5 pr. cent I Ans. g 11 12 cts, \^%m^ 

64 58 • 

5 



4 me» =t f 

I tno. = J 
10 days = 4 



32290 
3 



Interest for one year in cents* 



96870 for 3 years. 

10763 for 4 months. 

2690 for 1 month. 

896 for 10 days. 



Ans. g 11,12,19 = 1112 cts. or g 11, IS c.^l^^r 

2. What is the interest of g 325 41 cts. for 3 years, and 4 
months, at 5 pr. cent t Ans. g 54 23 cts. 5 m. 

3. What will 3000/. amount to in 12 years and 10 months, at 
Spr. cent? Ans. 5310/. 

4. What will g 730 amount to at 6 pr. cent, in 5 years 7 



months and 12 days J 



Ans. g 975 99 cts. 



AlUfriMEtlCKi 



INSURANCE, COMMISSION, amd BROKERAGE* 

Are allowances to Insurers, Factors, and Brokers, at a stipu- 
lated rate pr. cent, as a premium for their services* 

The same rules used in simple interest, apply to each of these 
cases. ' 

1 . What is the commission oh £287 10 s. at 3j pr. cent ? 

Ans. £10 1 s. S d. 

2. A Broker sells goods for me to {^amount of £2575 \T s. 
6 d. what is the brokerage at 4 s. pr. cwit ? Ans. £5 3 s. 0$. 

3. What is the insurance of a house, trained at 8 1853, at TS 
cts. pr. cent ? Ans. 8 13 89£ cfcu 



DISCOUNT. 

Discount is an allowance made for the payment of any sum 
of money before it becomes due ; and is the difference between 
that sum due some time hence, and its present worth. 

The fireaent worth of any sum, or debt, due some time hence f 
is such a sum, as, if put to interest, would in that time and at 
that rate pr. cent, for which the discount is to be made, amount 
to the sum, or debt then due. 

What remains after the discount is deducted, is the firescnt 
voorth. 

RULE, 
As the amount of 100/. or 100 dols. at the given rate and time : 
is to the interest of 100 at the same rate and time, so is the giv- 
#o sum to the discount. 

Subtract the discount from the whole debt, and Ihe remainder 
will be the present worth. 

Or ; as the amount of 100, is to 100, so is the given sum to 
the present worth. 

PROOF. 
Find the amount of the present worth for the time and rate 
proposed} which must equal the given sum, or debt. 

example. 
What must be discounted for the ready payment of 100 dols; 
due a year hence, at 6 pr. cent pr. annum ? 
As 106 : 6 : : 100 : 5 66 Ans. 
100,00 years sum. 
5,66 discount. 

$ 94,34 the present worth. 



If LOGARITHMIC* 

2. What sum in ready money, will discharge a debt of £9$Si 
tfitc * ydar an<j 8 months hence, at 6 pr. cent? r 

£ £ ££.*•*- 

As 110: 100 : ; 925 : 840 18 2 Ans. 

3. What is the present worth of 600 dols. due 4 years hence, 
at5pr. cent? Ans. 8500 

4. *What is the present worth of £*100, one quarter due in 
3 months, and the remaining 3 quarters, in 5 months, discount 
7 pr. cent ? Ans. £97 8 s. 10 d. + 

5. What is the difference between the it terest of 81904* at 
5 pr. cent pr. annum, £jf 8 years, and the discount of the same; 
for the same time and rate ? Ans. 8 137 60 cts. 



EQUATION OF PAYMENTS. 

Is finding the equated time, to pay at once, several debts due 
at different times, so that no loss shall be sustained by either 
* party. 

RULE. 
Multiply each payment by its time, add the several products 
together, and divide the sum by the whole debt ; the quotient 
will be the answer. 

PROOF. 
The interest of the sum, payable at the' equated time, will 
equal the interest of the several payments. ' 

examples. 
1. A ewes B. 8380, to be paid as follows, viz. 100 in six 
months, 120 in 7 months, and 160 in 10 months ; What is the 
equated time for the payment of the whole debt ? 
100 V 6= 600 
120 X 7= 840 
160 X 10= 1600 

380 )3040( 8 months. Ans. 

( . 3. The firm of B. & C. owe to the firm of B."& Co. the sum 
ftf 8 300; payments as follows: 100 in 3 months, 100 in 4 
months, and 100 in 6 months; required the equated time for 
the payment of the whole debt ? Ans. 4 \ months. 

Note. When Sundry sums are to be paid at different times, 
find the rebate, or present worth of each payment separately 
then add them into one sum. 



ARITHMETIC*. %fi, 

3. P. owes C. £420, which will be due 6 months hence, but P. 
is willing to pay £60 «o«>, provided he can have the rest re- 
main unpaid, a longer time than 6 months ; when must it be 
jaid ? Ans 7 months^ 



FELLOWSHIP. 

Fellowship, is a rule, by which Merchants, Sec. trading in 
jpmpany with a joint stock, are enabled to ascertain each per- 
sons particular share of the gain, or loss, in proportion to his 
share in the joint*tock. 

Fellowship is either single, or compound. 
Single Fellowship. 
Is when the several stocks in company are considered with- 
out regard to time. 

RULE. 
As the whole stock is to each mans share in sibek, so is the 
whole gain, or loss, to his share of the gain, or loss. 

PROOF. 
The sum of the several shares must equal the gain, or loss. 

EXAMPLES. 

1. A. B. and C. put in stock, and gain 800; A's stock was 
1200, B. 4800, and C. 2000 : What was each mans share of the 
gain ? 



A. 1200 
3. 4800 
C. 2000 



v****' 



8000 
* f 1200") fl2GY share of A. 

As 800: i 4800 V: : 800: i 480 i — ^- B. An*V : ' 

^2000 J 4260 J — t— • C. 

2. D. E. and F. trading, gained 120/. ; D's stock was 140K 
E's S00,and F's 160 4 W,hat was each mans share of the gain. 
« • • Ans. D's, 28/. E's, 60/. and FV32/1 

3.' Four men* trading with a stock of g 2400, and gain in tf 
years, twice as much, and g *60 more ; A's stock was 400, B'tr 
740, C's 820 ; what was D's stock, and how much did each marf 
gain. ■ _; 

Note. By this rule,* also, a bankrupt estate, may be divided 
among his creditors. 



N 



TSt iiOGARITHMlCK 

TVs stock, £ 440 cts. grills." 

A. gained R826 66 6 

Ans. -^B. gained £1529 33 3 

C. gained 81694T 66 6 

D. gained £909 S3 3 



COMPOUND FELLOWSHIP* x 

Is when tbe respective stocks of several partners in company 
are considered with time. 

RULE. 

Multiply each mans stock by its time, and add (he several 
products together, then 

As the sum of the products is to each particular product ; so 
is the whole gain, or loss, to its share of the gain, or loss. 

EXAMPLES. 

1 . Three Merchants trade together. A's stock is , 1 20/, for 9 
months, B's 100/. for 16 months, G's 100/. for 14 months, and 
they gain 100/: What is. each, mans share. 

A'sstock 120 X 9= 1080 : \.,- ; 
B's — — 100 X 16 .= . 1600 
G's ~- . lOO^X 14 =? 1400 

4080 sum. 

f 1080") f 26/. 9*. 4rf. i A. share. 

As 4080 : -] 1600 I : : 100/."" < 39/. 4*. 3d. J B. share. 

(^ 1400 J (_ 34/. 6*. 3d. £ C. share. 

2. Three Merchants join in company. H. puts in 8 620 for 8 
months, L. 950/. for 1 1 months, and M. 8 730 for 13ynonths, and 
(hey gain 1 800 : What was each mans share ? 

f A's share 358 55 4^ 
♦ . f Ans. 4 B's .755 42 1J£J . 

'trr :'"■ IP's — -4j 686 02 f^V 

i. A." began. trade, January i*.l 818* with a capital of g 1000, 
S*id : on the first of March following, took in B. as a partner, with 
a^api,ulof 8 1500 y three months after which, they admit C. a* 
*t,ttiird partner, who brought into stock g 2800, and after trading, 
together till the first of the next year, they find their gain to be> 
$ 1776 50: How ought each oneishare in tl»e profit? 
^.^ns. ^3 8457,46^11 B's g 571 83££J G's « 747$ff- 



/ 
i 

ARITHMETICS 

i . _ 

ALLIGATION. 

Teaches how to mix several simples of different qualities, so 
that the composition shall be of a mean, or middle quality. It 
consists of two parts, Alligation Medial, and Alligation Alternate. 
ALLIGATION MEDIAL, 

Is the method of finding the mean rate, or price of the com- 
pound, by having the rates and quantities of the several simples 
.given. 

RULE. 

Multiply each quantity by its rate ; then divide the sum of the 
products by the sum of the quantities, the quotient will be the 
rate of the compound required. 

EXAMPLES. 

1. Suppose 15 bushels of rye, at 64 cts. pr. bushel, 18 bushels 
of corn, at 55 cts. pr. bushel, and 21 bushels of oats, at 28 cts. 
pr. bushel, were mixed, what is the value of the composition pr. 
bushel ? 

bu. eta, g cts. 

15 X 64 = 9 60 

18 X 55 = 9 90 

21 X 28 = 5 88 



54 ~ 25 38 = 47 cts. Ans. 

2. If 18 bushels of wheat, worth g 1 50 pr. bushel, be mixed 
with 12 bushels of rye, at g 1 25 pr. bushel, what is a bushel of 
this mixture worth I Ans. g l 40 

3. Suppose a Wine Merchant mixes together 73± gallons of 
wine, at g 2 16 pr. gallon, 5£ gallons, at g 2 pr. gallon, and 4£ 
gallons, at g 1 80 pr. gallon ; what will a gallon of this mixture 
be worth, supposing he should accidentally spill a quart cf wa- 
ter into it ? Ans gl 9S-J 

4. A Goldsmith melted together 5 lb. of gold of 22 ci^ats 
fine, 22 lb. of 21 carats fine, and 1. lb. of alloy ; what is the qual- 
ity of the mass? Ans. 19 carats fine. 

ALLIGATION ALTERNATE, 
Is the method of finding what quantity of any number of in/- 
gredients, whose rates are given, will compose a mixture of a 
given rate : so that it is the reverse of Alligation Medial, and 
may be proved by it. 

RULE. 
1. Write the several rates, or prices of the simples, in a, 
column under each other, and the mean rate, c»r piiCv yf ihe 
whole mixture, at the left hand. 

iO 



I 



LOGARITHMICK 

2. 'Connect, or link the price of each simples or ingredient, 
which is less than the mean rate, or price of the whole mixture, 
with one, or any number of those, which are greater than the 
mean rate, and each greater rate, or price with one, or any num- 
ber of the less. 

3. Write the difference between the mean firice, (or mixture 
rate) and that of each of the simples, opposite to the rates with 
which they are connected. 

4. Then if only one difference stand against any rate, it will 
be the quantity belonging to that rate; but if there be several, 
their sum will be the quantity. 

EXAMPLES. 

1. A Merchant would mix wine at 14s. 19s. 15s. and 22s. pr. 
gallon, so that the mixture may be worth 18s the gallon ; what 
quantity of each mu3t be taken ? 

M«n !"!*;-)?"'«• or 



— ^. , 0lis!£*"»^:g|: 

1 22 — > 4 at 22s. 22— > 4 



14 » 1 + 4 



5 at 14s. 
1 at 15s. 
7 at 19s. 
4 at 22s. 



2. How much wine at 6s. pr. gallon, and at 4s. pr. gallon, 
must be mixed together, that the, composition may be worth 5s. 
pr. gallon ? Ans. 12 gallons of each. 

3. A Merchant would mix several kinds of spirits together; 
some at 7s. some at 10s. some at 5s. and some at 13s. pr. gallon ; 
how much of each sort must the mixture contain, so that a gal- 
lon of it shall be worth 9s. 2d ? 

Ans. 4 gal. at 5s. 1 gal. at 7s. 2 gal. at 10s, and 4 gal. at 13s. 

4. How much grain at 2s. 6d. 3s. 8d. 4s. and 4s. 8d. pr. bush, 
must be mixed together, that the compound may be worth 3s. 
lOd. pr. bushel ? 

Ans. 12 at 2s. 6d. 12 at 3s. 8d. 18 at 4s. and 18 at 4s. 8d. 

5. How much water at pr. gallon, may be mixed with li- 
quors at Is. 7s. and 8s. pr. gallon, so that a gallon of the mix- 
ture may be sold for 5s. pr. gallon ? 

Ans. 5 of water, 5 at Is. 9 at 7s. and 9 at 8s. 



* Note. By connecting the less rate with the greater, and 
placing the differences between them, and the mean rate alter- 
nately ; the quantities resulting, are such, that there must be 
precisely as much gained by one quantity, as is lost by the 
other; therefore the gain and loss upon the whole are equal, 
and arc exactly the proposed rate. It is also obvious, that ques- 
tions under this rule, admit of answers differing, ad infinetum ; 
for having found one answer, we ma\ find as many more as we 
please, by only multiplying, or dividing each of the quantities 
found by 2, 3, 4, &c. 



ARITHMETICS 75 



TARE AND TRETT. 

Tare and Trett are allowances made to the buyer, on some 
particular commodities. 

Tare is the weight of the barrel, box, bag, or whatever con- 
tains the articles. 

Trett is an allowance of 4 lb. in every 104 lb. for waste, 
dust, &c. 

Gross, is the weight of the goods together with the barrel, 
box, bag, or whatever contains them. 

When the tare is deducted from the gross, it leaves what is 
called the suttle. 

Neat, the weight of the goods, after all allowances are 
made. 

CASE I. 

When the tare is a certain weight fir. box^ barrel^ Istc. 

rule. 

Multiply the number of boxes, or barrels, Sco. by the tare, and 
subtract the product from the gross, the remainder is the neat 
weight required. 

EXAMPLES. 

1. In 7fraits of raisins, each Weighing 5 cwt. 2 qr. 5lb. gross, 
tare 231b. pr. frail; how much neat I 
23X7 = 1 cwt. lqr.2llb. 

S> 2, 5 What is the neat weight of 14 hogs- 

T heads of tobacco ; each weighing 

— — -r- 5 cwt 2 qr. 17 lb. gross, tare 100 pr. 

38, 3, 7 gross, hogshead, Ans. 66 cwt. 2 qr. 14 lb. 
1, 1, 21 tare. 



S7, 1, 14 the answer. 



CASE II. 
When the tare is a certain weight fir. ciut. 
RULE. 
Divide the gross weight by the aliquot parts of a cwt., con- 
tained in the tare, and subtract the quotient from the gross ; thet 
remainder is the neat weight. 



7£ LOGARITHMIC* 

EXAMPLES. 

1. Gross 173 cwt. 3 qr. 17 lb. tare 16lb. pr. cwt how mutch 
peat ? * 

cwt. gf. lb. 
173 3 17 grow 

"U lb. = | 21 2 26 2. What is the neat weight of 7 barrels 

2 lb. = 4 3 11 of potash, each weighing 201 lb. gross, 

— — — tare 101b. pr. cwt. I Ans. lS8lb.6©z. 
24 3 9 



149 8 Ans. 



3. In 25 barrels of figs, each 2 cwt. 1 qr. gross, tare 16 pr* 
Cwt. ; how much neat ? Ans. 48 cwt. 24ib* 

CASE III. 

When Trett i* allowed with Tare. 

RULE. 

Divide the suttle by 26, and the quotient will be the Trett, 

which subtract from the suttle, the remainder is the net weight. 

EXAMPLES. 

1. In 9 cwt. 2 qr. 17 lb. gross, tare 37 lb. and trett as usual f- 
how much neat ? 

/*. 

2. In 7 casks of primes, each 
weighing 3 cwt. 1 qr. 5 lb. gross, 
tara \7\ lb. pr. cwt. and trett as 
usual ; how much neat ? 
Ans. 18 cwt. 2 qr. 25 lb. 

8 3 25 Answer. 

CASE IV. 
When Tare Trett and Cloffare allowed. 
RULE. 
Deduct the tare and trett as before, divide the suttle by 16$, 
the quotient will be the cloff, which subtract from the suttle ; 
the remainder is the neat weight.* 



cwt. 
9 



gr. lb. 
2 17 
1 9 


gross* 
tare. 


► J9 


1 8 
1 It 


suttle. 
trett. 



ARITHMETICS 7t 



• EXAMPLES. 

1. What is the neat weight of a hhd. of tobacco, "weighing 
15 cwt. 3. qr. 201b. gross, tare 7 lb. pr. cwt. and trett and cloff 
as usual ? 

cwt. or. lb. 

2. In 19 chests of sugar, each 
containing 13 cwt. 1 qr. 171b. 
gross, tare 131b. pr. cwt. and 
trett and cloff as usual ; — how 
much neat ? and what is the 
value 5^d. pr. lb. ? 
Ans. 215 cwt. 171b. and value 
£577 6S. Si 



cwt. qr. 
15 3 
71b^ 3 

26)14 3 
2 


lb. 

20 gross. 

27 ure. 

21 

18 trett 


168)U 1 


13 suttle 
9 cloff 


14 1 


4 Ans 



COMPOUND PROPORTION, OR DUBLE RULE 
OF THREE, 

Teaches to solve, at once, such questions, as require two, or 
more statings in simple proportion, whether Direct, or Inverse. 

In this rule, their are always five terms given, to find a sixth. 
The three first terms of which, are a supposition, the two last a 
demand. 

RULE. 

In stating the question, place the terms of supposition, so that 
the principal cause of loss, gain, or action, possess the first 
place ; that which signifies time, distance of place, in the second 
place, and the remaining term in the third place. Place the 
terms of demand under those of the same kind, in the supposi- 
tion. 

If the blank place, or term sought, fall under the third term, 
the proportion is direct ; then multiply the first and second 

Note. The following method of stating compound proportion 
fa, by some, prefered, 

1. Place that number, or term, which is of the same name, or 
kind with the answer, in the third place. 

2. Then take one term from the supposition, attd one from 
the demand, both of the same name, or kind, and place thera 
with the third term. 

K 3. Then proceed in the^ same manner with the two remaining 
terms. 



78 LOGARITHMIC* 

terms together for a divisor, and the other three for a dividend : 
but if the blank fall under the first, or second term, the propor- 
tion is Inverse; then multiply the third and fourth term togeth- 
er for a divisor, and the other three for a dividend ; and the 
quotient will be the answer. 

4. Reduce the similar terms to the same denomination, if 
necessary. 

5. Multiply the terms in the second and third place together, 
and divide their product by the product of those in the first 
place j — the quotient will be the answer. 

EXAMPLE* 

If 7 men can build 36 rods of wall in 5 days, how many rods 
can 20 men build in 14 days ? 

. men 7 : 20 : : 36 
days. 3 : 14 

21 280 

2 \) 10080(480 
84 

168 
16-8 

00 
In compound proportion, therfe are always five numbers given 
to find a sixth, which multiplied into the product of the two first, 
shall be equal to the product of the other three. And it maybe 
shown in compound, as in simple proportion, that the product of 
the extremes, is equal to the product of the means ; thus 
2X3: 12:: 3X4: 24; here the extremes are 2X3 and 24, 
and the means are 12 and 3X4; now the product of 2 X 3 X 24 
the extremes, is 144 ; and the product of 12 X 3 X 4 the means, 
is also 144; whence it is evident that dividing either the pro- 
duct of the three extremes, or mean» % by any two of the means 
gives the other mean ; thus, 144 divided by the product of the 
two means 3 X 4, gives 1 2 for the other mean ; and it is also 
manifest, that by dividing the product of the three extremes, or 
incantty by the product of any two of the extremes, the other 
extreme is obtained ; thus dividing 144 the product of the 
means, by 2 X 3 the product of two of the extremes, gives 24 
for the other extreme : hence the propriety of the rule in mul- 
tiplying the Sd. 4th. and 5th. terms, or means together, and di- 
viding by the product of the two first terms, or extremes, ta 
obtain the other extreme. 



ARITHMETIC*. $» 

EXAMPLES. % 

If 7 men can build 36 rods of wall in 3 days, how many rods 
can 20 men build in 14 days ? 

7:3:; 36 terms of supposition. 
20 14 terms of demand. 

56 



If 150/. in 12 months, gain 9/. in what 
time will 450/. gain 54/. ? 

150 : 12 : : 9 

450 : : 54 Ans. 2 years. 
7X3 = 21)10080(480 rods. Ans 

BY LOGARITHMS. 
In compound, as in simple proportion, the term required may 
he found by logarithms, if we substitute addition for multiplica- 
tion, and subtraction for division. 

RULE. 
Add together the logarithms of those terms which in com- 
mon arithmetick are to be multiplied together for a dividend, 
and from the sum,, subtract the sum of the logarithms answer- 
ing to these terms, which in common arithmetick, are multipli- 
ed for a divisor ; and the remainder will be the logarithm of the 
answer. Or more fully ; find the arithmetical complement of 
the logarithms to be subtracted, and then add all the terms to- 
gether ; the sum will be the logarithm of the answer. 

examples. 
2. If 4 men in 12 days, mow 48 acres ; how many acres can 
8 men mow, in 1 6 days . ? 

Two first terms j 1 2 

Third term 8 

Fourth and fifth terms < lfi 

* Term required 128 Ans. 2. 1072100 



a. c. 


9. 


3979400 


a. c. 


8. 


9208188 




0. 


9030900 




1. 


6812412 




1. 


2041200 



* It must be observed, that each arithmetical compliment in- 
creases the index of the logarithm by 10 ; as often therefore, 
as the a. c. is used, the index of the sum of the logarithms, 
must be made less by so many tens. 



& LOGAWTHMICK 

1. If 10 bushels of oats, be sufficient for 18 borses 20 days* 
.How many bushels will serve 60 horses 36 days ? 
T60 log. 1. 7781513 
Three last terms i 36 l. 5563025 

(.10 1. 0000000 

Sum of Log. 4. 3344538 * 

L*g. to be subtracted JJJ log 



1. 2552725 
1. 3010300 



2. 5563025 

Sum of the Log's, three last terms 4. 3344539 
do. of the two first 2. 5563025 



Term required 60 =* 1 . 778 15)3 



3. What principal will gain £*2&2 10s. in 7 years, &\£5 pr«- 
cent I Ans. £750 

4. If the interest of 365 dollars for 3 years and 9 months, be 
g 82 13 cts. what will be the interest of g 8 940 for two years 
and 6 months ? Ans. 1 340 

5. If/100 in 12 months, gain £6 interest; how much will 
£"75 gain in 9 months ? Ans. £ 3 7s. 6d. 

6. If jf 16 18s. be the wages of 16 men, for 8 days ; what sum 
will 32 men earn in 24 days ? Ans. £ 01 8s. 

7. If £ 100 in 12 months, gain £7 interest ; what principal 
will gain £3 18s. 9d. in 9 months i Ans. £ 75 

8. If 2001b. be carried 40 miles, for 40 cents; how far may 
-202001b. be carried for g 60 60 cts. Ans. 60 miles. 



COMPOUND INTEREST. 

In calculating compound interest, the amount for the first year* 
Is made the principal for the second year ; and the amount for 
the second, the principal for the third, Sec. 

As the Logarithmic ka I method of computing compound inter* 
est, is by mutch the most cxpediiious, it is thought unnecessary 
to subjoin the old one in this place. Therefore, 

To calculate Compound Interest by Logarithms. 
RULE. 

1. Find the amount of 1 dollar for 1 year; multiply its loga- 
rithm by the number qf years, and to the product add the loga* 



abjthmetick; jj, : 

rithm of th£ principal. The sum will he tfye ^arkhm of tic 
Amount for the given time. 

2. From the ampunt subtract the principal, and the remainder 
will be the interest. 

EXAMPLES. 

1. What is the amount of 20 dollars, at 6 pr. cefit compound 
interest, for 100 years ? 
Amount of 1 dollar for 1 year = 81 06 log. 0.0255059 
Multiply by the time \<K) 

2.5305900 
Add log. of S 20, given principal 1.3010300 



Amount required 8 6786 3.8316200 

2. What is the amount of 425 dollars, at S pr. cent compound 
interest, for 4 years ? 

Amount of 1 dollar for 1 year == g 1 05 log. 0.021 1893 
Multiply by the time 4 



0.0847572 
Add log. of 425 given principal 2.6283889 

Amount 8516 59 3.7131461 



Note 1 . If the the interest becomes due semianually^ or guar* 
ttrly ; find the amouaf of one dollar, for the half-year or quar- 
ter, and multiply the logarithm, by the number of half-years or 
quarters in the given time. 

Note 2. As Simple Interest is performed by a rank of numbers, 
arithmetically proportional, so it may be shown, that Compound 
Interest is performed by a rank of numbers geometrically pro- 
portional. 

And it is a principle in Mathematicks, that, if three num- 
bers be in geometrical proportion, the product of the two ex- 
tremes, is equal to the square of the mean. (See Euclid's Eli-> 
ments, 20th prop. 7th book.) And on the contrary, if the rec- 
tangle contained by the extremes of any three numbers, be equal 
to the square of the mean, then those three numbers are in. 
geometrical proportion. 

Now if 3 dollars be the compound interest of 8 100 for \ year, 
or 6 months, then these three numbers 100,103, 106, should be in 
geometrical proportion ; but it may be proved by the aforesaid 
proposition, they are not; for the reetangle of 100 into 106 is 
but 10600, and the square of the mean 103, is 10609, which is 
greater than the product of the two extremes. But the square 
root of 10600 will be found to be 102,956 : so that the true pro- 
portional interest of g 100, for i year, is but g 2 95 cts. 6 m. t 
11 



A» LOGARITHMIC*: 

3. What i* tte anuMintof 1000 dollars, a* 6 pr. cent 
pound interest) for 10 yean* ? 4ns. $1790 

4. Required the amount xrfiOD cblla», at 6 pr. cent compound 
interest, for 3 years? Ans. 8 119 10 

5. What will 1000 dollars amount to at 7 pr. cent, compound 
interest, in 4 year? ? Aas. 8 1310 80 

6. What is the compound interest of 876 dollars 90 cts. at 6 
j>r. cent pr. annum, for 3 yeai? and 6 months? Ans. $ 198 83-f- 

7. What will 100 dollars amount to in 3 years, at 6 pr. cent 
compound interest} allowing that it becomes due semiannually £ 

Ans. 2 127 054- 

8. What is the amount of 400 dollars, at 5 pr. cent com; 
pound interest, for 1 year, payable quarterly ? Ans. 420 37 

9. What is the amount of 1 cent, at 4 pr. cent compound in* 
fewest, in 500 years ? 

Amount of 1 dollar for 1 year== 8 1 06 log. 0,0253059 
Multiply by the time 500 

I I I II MMMM 

12.6529500 
Add log. of principal 8.0 01 — 2.0000000 



Amount 8 44,973,000,000 IQ;6329$6b 



INVOLUTION. 

A Power is a number produced by multiplying &y given 
number continually by ktelf a certain number of times. 

The number denoting the power, is called the Index* jonr 
Exponent of that power* 

To rai&c a ghten number, woe hattt the foltevtins 
RULE. 

Multiply the given number, or first power, continually by it? 
self, tin the number of multiplications be 1 less than the index 
of the power to be found, and the last product will be the power 
required. 

Note. Powers are commonly denoted by writing their indices 
above the first power; as follows. 

2X2=4, the 2d power, or square of 2, or 2*. 
£ X 2 X 2 a 8, the 3d power, or cube of 2, or 2', 
t X 2 X 2 X 2 = 16, or hiquadfete of h <« 2 ** * c - 



ARITHMETICS u 

XXAtfMSS* 

h Let it be required to raise 45 to its cube, or third power ? 

45 



10125 
8100 

Ans. the 3d power, 91125 or cube of 45 
2. What is the square of 3758 ? Ans. 14122564 

3., What is the cub* of 327 ? Ans. 3-4965783 

4. What is the bftfuadfltte, or fourth power qf 376 ? 

Ans. 19987173376: 

5. What is the fifth power of -029 ? Ans. -000000020*1 1 149 

6. What is the sixth power of 48 ? Ans. 12330&9O464 

7. Required the seventh power of 7 ? Ans* 823543 

. EVOLUTION- 

Is that, by which we extract the roots of number*; or find a 
radical quantity, which multiplied into itself a certain number of 
times will produce the given power. 

1*0 EXTRACT THE SQUARM ROOT. 

RULE* 

1. Having distinguished the given number in periods of two 
figures each, beginning at the place of units, find the greatest 
square number in the first, or left hand period, place the root of 
it at the right hand of the given number, (in the manner oi a 
quotient figure ia division,) for the first figure of the root, the 
square of which subtract from the first period, and to the re- 
mainder bring down the next period for a dividend. 

2. Pkcte the double of the root, already found, on the left of 
the dividend for a, divisor. . 



Note. Roots are sometimes denoted by writing </ before the 
power, with the index, of the root against it. Thus the third 
r,oot of 80 is ^ i B0, an4 the second root of 80 is V 80 > the index 
T y though omitted, is always to be understood, when the radical 
sifep is written without a numeral index, 



** 



LOGARITHMICK 



3. Consider what figure must be annexed to the divisor, so 
that if the result be multiplied by it, the product may be equal 
to? or next teas than the dividend, and it will be the second fig- 
ure of the root 

4. Find a divisor as before, by doubling the figures already 
in the root ; and from these find the next figure of the root, as 
in the last article ; and so on through all the periods to the last. 

/ EXAMPLES. 

1. What is the square root 2. 

of 14122564? of 

14* 12*25*64 | 3 the root' 
9 



What is the square root 

5499025 ? 

5.49*90-25 I 2345 root. 

4 



67 I 512 
469 



43 I 149 
129 



745 I 4325 
3725 

7508 I 60064 
60064 



464 I 2090 
1856 



4685 J 23425 
23425 



remains. 

3; What is the square root of 10342656 ? 

4. What is the square root of 43264 ? 

5. What is the square root of 451584 ? * 

6. What is the square root of 2985984 ? 
T. What is the square root of 
$. What is the square root of 



998001 



remains. 
Ans. 3216 
Ans. 208 
Ans. 672 
Ans. 1728 
Ans. 999 



9*4,5192360241 ? 

Ans. 31,05671 

9. What is the square root of 103089* 198,4001 ? 

Ans. 32107,51 

1 0. * What is the square root of 1 60 ? Ans. 1 2,649 1 1 4- 

1 1 . What is the square root of 2 ? Ans. 1 ,4 1 42 1 35 6237 +• 

12. What is the square root of 10 ? Ans. 3.162277 &c. 



* Note. When the given number is a surd ; that is, when its 
root cannot be found exactly, without a remainder, the evolu- 
tion may be carried on, until we obtain a root, sufficiently near 
the truth, by annexing cyphers to the remainder, and proceed- 
ing as in rational numbers. In the 10th example ; although 
12.6491 1, is not the exact root of 160, yet if it be multiplied by 
itself, the product will be 159,9999837921, which is not two 
parts, of which 10000 taskc an unit, wide jof thetrfltlu, 



ARITHMETIC*. 8* 



TO EXTRACT THE CUBE ROOT. 

RULE. 

1. Having distinguished the given nuirfber into period* of 
three figures, find the nearest less cube' in the first period, set 
its root in the quotient, and subtract the said cube from the 
first period; to the remainder bring down the second period* 
and call this the Resolvend. 

2. To three times the square of the root, just found, add three 
times the root itself, setting this one place farther to the right 
than the former, and call this sum the Divisor. Then divide 
the resolvend, excepting the right hand figure, by the divisor, 
for the next figure of the root, which annex to the former ; 
calling this last figure e> and the part of the root before found, 
call a. 

3. Add together these three products, namely, three times 
the square of a multiplied by e, three times a multiplied by the 
square of ?, and the cube of e\ setting each of them one place 
farther towards the right than the former, and, call the sum the 
Subtrahend x which must not exceed the resolvend ; if it 
does, then make the last figure e less, and repeat the operation 
for finding the subtrahend. 

4. Subtract the subtrahend from the resolvend, and to the 
remainder bring down the next period of the given number for 
anew resolvend ; to which form a new divisor from the whole 
none root found ; and thence another figure of the root, as before 
and thus continue till the wh<#e is finfehed. 



•6 LOGARITHMICK 

EXAMPLES. 

H Required the cube root of 4360368248*7 > 

3X7* = t& | 436-036M4-2ST \ 7583 ttfe Mot: 
3X7 = 21 J 343 



l%\. 9msor. 1511 


93036 1st. resofvend. 
78875 subtract subtrahend* 


r 3X7*X5 735 
Add J3X7X5»=525 


I 4! 6 1824 2d. resolvend. 

1 36445 f 2 sub. 2d. subtrahend. 


* 5* = 125 
1st. 78875 


517312287 3d. resolvend. 
517312287 sub. 3d. sub'end. 


3X75 * » 1787& remains. 

3X75 gs 225 


2d. Diviso* IT8975 

f 3X75*X8 ^ 135000 
Add« 5X75X8* =*= 14400 
I 8* =- 512 


2d. Subtrahend 

3Xr58* =* 
3X75* 


13644512 

1725692 
2274 


Gd. Divisor. 

f 3X7582X3 = 

Add J 3X758X32 «= 

1 3' « 


17239194 

5171076 
20466 
27 


3d. Resolvcnd 


S1731228r 



The laborious operation of extracting the roots of higher 
pewers, is often so tedious, as to render it highly irksome and 
'forbidding to learners. But, as in Logarithms, addition is 
made to perform the office of multiplication, and subtraction 
that of division ; the labour of evolving roots is not only short- 
ened to a degree surpassing credence, but the whole h per- 
formed with incomparable facility and expedition. 



ARITHMETICS. 



87 



EVQUJTJON BY LOGARITHMS. 



Evolution is the opposite of involution. And it was shown 
in the introduction of logarithms, that quantities are involved, by 
multiplying their indices^ or logarithms. For the same reason, 
therefore, the roots of quantities may be, extracted, by dividing 
their indices, qr logarithms. 

To extract the root of any number by logarithm^. 
We have therefore, this general 

RULE. 
Divide the logarithm of the given quantity, by the number 
expressing the root to be found. 

examples. 
1. Required to find the cube root of the same number 
436036824287, by logarithms ? 

Numbers. Logarithms. 

Power 436026824297 3 j 11. 6395233 

Root 7583 3. 87984 ii 



2. What is the squre root of 92613 
Numb. 
Power 9201 
Root 21 

3/ What is the square root of 9801 ? 
Numb, 
Power 98#l 
Root 99 

4. What is the square rpot of 365 ? 
Numb. 
Power 365 
Root 19.10498* 



5. Required the cube root of 12345 I 
Numb. 
Power 12345 
Root 23.11162 



«. What is the 10th root of 2 ? 
Numb. 
Power 2 

Rbbt 1.000121 



Log. 

3 J 3. 9669579 
1, 3222193 



Log. 
2 J 3. 9913704 
1. 9956352 



Log. 
2 | 2. ,5622929 
1. 2811465 



Log. 
3 I 4. 0914911 
1. 3638304 



Log. 
10 | 0. 3010300 
0. OSOlOof), 



18 



LOGARITHMIC* 



7, Required the 10th root of 6948 ? 
Numb. 
Power 6948 
Root 2.422 



8. What is loeth root of 983 ? 
Numb. 
Power 983 
Root 1.071 



10 [ 3. 6418598 
0- 3841859 



Log. 
£00 | 2. 9925535 
0. 0992553 



9. Required the 365th root of 1.045 I 

Numb. Log. 

Power 1.045 365 | 0. 0191163 

Root 1.0Q121 0. 0000524 



♦10. Required to find the 10000th root of 49680000 ? 
Numb. Log. 

Power 49680000 10000 | 7< 6961816 

Root 1.0017899 0. 0006961 



The Logarithms of Povters given, to find their roots. 
X. Required the square root of 6561 

Powers. Logarithms* Roots. 
6561 3. 8169700 Ans. 81 


2. Of 


4096 


3. 6123599 


Ans. 64 


3. Of 


15625 


4. 1938200 


Ans. 125 


4. Of 


46656 


4. 6689076 


Ans. 216 


5. Of 


M7649 


5. 0705882 


Ans. 34S 


6. Of 


262144 


5. 4185400 


Ans. 512 


7. Of 


531441 


5. 7254550 


Ans. 729 


8. Of 


1679616 


6. 2048674 


Ans. 1266 


9> Of 


5764801 


6. 7607844 


Ans. 240 1 


to. Of 


43046721 


7. 6339400 


Ans. 6561 


Required the cube root 
Powers* 
1. Of 1728 


of the following \ 
Logarithms. 

3. 2375437 


numbers. 

Roots. 
Ans. 12 


2. Of 


8000 


3. 9030900 


Ans. 20 


3. Of 


15625 


4. 1938200 


Ans. 25 


4, Of 


19683 


4. 2940914 


Ans. 27 


5. Of 


10077696 


7. 0033614 


Ans. 216 


6. Of 


244140625 


8. 3876400 


Ans. 625 


7* Of 


68719476736 


10. 8370797 


Ans. 4096 



♦Note. We have here an instaqpe of the great rapidity with 
which arithmetical operations are performed by the use of Lqx>- 

ARITmfS. 



ARVTKMETJCR. W 

«. Ui yyvo'00029999 11. 9998698 Ans* 9999 

9. Of 205884571094649 14. 3136375 Ans. 59049 

20. What is the fourth root pf 19987173376 ? 

Log. = 10, 3007512 sb Ans. 376 the root. 

11. What is the fifth rpot of 507682821 106715625 ? 

Log. « 17. 48810?0 « Ans. 3145 

12. What is the sixth root of 43572838 1009267809889764416? 
Log. = 26. 6392164-7-6 = 4. 4398694 «= ${7534 the Aos. 

1 3. What is the seventh root of 

34487717467307513182492153794673? 
Log. = 31. 5376642 --7 = 4. 5053806 = 32017 Ans. 

14. What is the eighth root of 

1121016231320476236246497942460481? 
Log. = 33. 0496120—8 = 4. 1312015 = 13527 Ans. 

Nothing can be more easy, than to extract the roots of powers, 
to which the. logarithms are givet, or may easily be fou*d. But a 
difficulty may sometimes arise, in the learner'sjfotfiftj' the exact 
logarithm to a proposed number, that is much greater than any 
in the Tabus; yet a very superficial attention) to the nature of 
logarithms, will readily suggest a solution of this seeming diffi- 
culty ; for, as adding the logarithms of several numbers, is 
equivolent to multiplying by the same numbers, and subtracting 
the logarithm of numbers, the same, as dividing by those num- 
bers; therefore, 

To find the logarithm to a proposed J&umber greater than any 
in the Table. 

RULE. 

Resolve the given quantity into such factors, as will consti- 
tute it within the limits of the table, add together the logarithms 
of these factors, and the sum will be the logarithm to the pro- 
posed number. 

3P*AMPJU*S. 

I. Required the logarithm of the natural nunaber, 11042f 
Here it is evident, that dividing the. given quantity by two, will 
constitute it within the limits of the Table ; as follows, 
Factors, 
3)11042(5521 Log. 3. 7420177 

2 Log. 0. SO 10300 

Given number. 11042 required Log. 4, -0430477 
12 



9ft 



LOGARITHMIC* 



2. Find the logarithm of 15378* 
Factors. 

3)15378(5126 



Ltfg. 3. 7097785? 
Log. 0. 4771213 



Given number I5S78 required L<%. 4. 1868999 

3. Required to'fl^d the logarithm of 17304 
Factors. 





4)17304(4326 

4 


Log. 
Log. 


, 3. 6360805 
0. 6020600 




Given number. 


17304 


required Log. 


4. 2381405 


4. 

5. 


What is the logarithm 
Of 19505 ? 
Of 25596 ? * 


Ans. 
Ans. 


Logarithms. 
4. 2923068 
4. 4081722 


6. 


Of 39126? 




Ans. 


4. 5924655 


7. 


Of 57320 ? 




Ans. 


4. 7583062 


& 


Of 71464 ? 




Ans. 


4. 8540873 


9. 


Of 89897 ? 




Ans. 


4. 9513229 


40. 


Of 119844? 




Ans. 


5. 0786162 


11. 


Of 217975 ? 




Ans. 


5. 3384067 


12. 


Of 3089725? 




Ans. 


6. 4899199 



CASE II. 
When there are cyphers on the right hand of the given num- 
ber. 

RULE. 
Find a logarithm to the significant figures, as before, and in- 
crease the index by as many units, as there are cyphers on the 
right of the givep number. 

EXAMPLES. 

1. What is the logarithm of 57640 ? 

The logarithm of 5764 is 3. 7507240 ; pnd increasing the 

index 3, by 1 j we shall have 4. 

57640 

2. Required the logarithm 

Of 586400 

3. Of 6495000 

4. Of 72970000 
5- Of 910100000 
6. Of 44.973.00^000 



7607240 for the logarithm of 



Ans. 
Ans. 
Ans. 
Ans, 

Ans. 



5. 
6. 
7. 
8. 

19- 



7681940 
8125792 
8632634 
959089 1 
6529:5 GO 



AftlTHMEtlpK. 91 



JPlRjtCTICAL QUESTIONS W EVOLUTION AMI* 
INVOLUTION. 

BROBLEM I. 

To find & mean proportional between two numbers. 

RULE. 
Add together the logarithms of the given numbers, divide 
the sum by 2 ; the quotient will be the logarithm of the mean- 
proportional required. 

1. Required the mean proportional between 45 and 180.. 

1. 6532125 
Thus \ 180= 2. 2552725 



f 45 = 



2)3. 9084850 
Mean proportional required 90 e= 1. 9542425 

2. Required a mean proportional to the numbers 64 and 256 

Ans. 128 
PROBLEM II. 
Any number of soldiers being given, to place them in a square 
Battalia of men. 

RULE. 
Divide the logarithm of the given number by 2 ; the quotient 
will be the logarithm of the answer. 

3. Let 9216 men be ordered to form a square battalia ; how- 
many must stand in rank and file ? 

9216 2)3. 9645425 Log. 

Ans. 96 1. 9822712 

4. How many must stand in rank and file, so that 5625 men 
shall compose a square ? Ans. 75 

5. Let 8450 men be so formed, as that the number in rank 
may be double the number in file. 

8^50^-2= 4225 = log. 3. 6258267 ~ 2 = 1. 8129J33 = 6s 
2)8450 

4225 log. 2)3. 6258267 

Ans. 65 in file * 1. 8129133 

Multiply 2 

And 130 in rank. 



Note. When the question requires double, trible, or quadru- 
ple, the number of men to stand in rank, as in file ; divide the 
logarithm of t, -f, £ &c. of the given number by 2, the quotient 
wll be the number in file, which double, triple, quadrupfo 
Sec. apd the product will be the. number in rank. 



i 



9& LOGARITHMIGK 

6. Required to set out 27648 fruit trees, so that the number 
in length, $hall bfe to the number in breadth as 3 is to 1 ; how,, 
must they be placed ? 

Ans. 288 in length, and 96 in breadth. 
PROBLEM III. 
Any two sides of a rightangled triangle being given, to find 
the other side. * 

Case 1. When the base and perpendicular are given, to find 
the hypothenuse. 

RULE, 
Add the squares of the two legs together, and extract the 
square root of the sum. 

7. A triangular piece of ground measures 30 rods on one side,, 
and 40 rods on another ; required the length of the remaining, 
or longest side. 

30 X 30 = $00 
40 X 40 ex 1600 

3500(50 hypothenuse, or 
25 longest side 

oo - 

8. Required the length of a brace in a building, so that the 
lower end of it shall be 8 feet, and the upper end, 6 feet from 
the right angle. Ans. 10 feet. 

9. Suppose the lower end of the brace to rest in a post 3 feet, 
and the upper end framed into a plate 2 feet 3 inches from the 
right angle : required its length. Ans. 3 feet 9 inches. 

10. What will be the length of a brace, when it is required 
that the distances from the right angle to either end, should be 
2 feet 6 inches, and 3 feet 4 inches ? Ans. 4 feet 2 inches. 

1 1 . Two stages start from the same- place ; one goes directly 
south at t^e rate of 9 miles an hour, for 3 hours, the oiher due 
west, for 4 hours, at the same rate ; in what time would they 
now meet, were their course turned directly towards each other, 
continuing at the same rate per hour ? Ans. 2-J- hours. 

12. Required the length of a scaling ladder, to reach the top 
of a wall, whose height is 28 feet, the breadth of the ditch be- 
fore it being 45 feet ? Ans. 53 feet. 



♦Note. The square of the hypothenuse, or the longest side 
of a rightangled triangle (by 47th. Theorem B. 1. Euc.) is 
equal to the sum of the squares of the other two sides ; and 
consequently the the, difference of the squares of the hypothe- 
nuse and either of the other sides is the square of the remain- 
ing side. 



ARITHMETICS. §* 

PROBLEM IV. 

CASE 2. 

The hypothenuse and one leg. being given, to find the other 
log. / 

RULE. 

Subtract the square of the given leg. from the square of the 
hypothenuse, and extract the square root of the difference. 

13. What is the perpendicular of a right angle triangle, whose 
base is 56 feet, and hypothenuse 65 ? Ans. 33 feet. 

14. What is the base of a right angled triangle, the hypothe- 
nuse being 159 feet, and the perpendicular 84 feet ? 

Ans. 135 feet. 

15. A line of 65 yards will reach from the top of a precipice, 
standing close by the side of a brook, to the opposite bank ; re- 
quired the breadth of the brook, the heighth of the precipice 
being 33 yards ? Ans. 56 feet. 

1 6. A ladder of 50 feet long, being placed in a street, reached 
a window 28 feet from the ground on one side ; and by turning 
the ladder over, without removing the foot, it touched a mould- 
ing 36 feet high on the other side ; required the breadth of the 
street. Ans. 76. 1233335 feet. 

17. Two ships sail from the same port; one, due east 84 
leagues, and the other, directly south 135 leagues: how far are 
they asunder I* Ans. 159 leagues. 

PROBLEM V. 
To find the circumference of a circle from its diameter. 

RULE. 
Multiply the diameter by 3 # 14159 

OR 

Multiply the diameter by 355, and divide the product by 113. 
Ex. 1. If the diameter of the earth be 7930 miles, what is 
the circumference ?• 3. 14159 X 7930 = 24913 miles. 

2. How many miles does the earth move, in revolving round 
the sun ; supposing the orbit to be a circle, whose diameter is 
190 million miles? Ans. 596,902,100. 

3. If the diameter of a wheel be Af feet ; what is the circum- 
ference? Ans. 14 feet 1-J- inches. 

* Note. The square toot may in the same manner be applied 
to navigation ; and when deprived of other means of solving 
problems of that nature, the following proportion will serve to 
find the course. 

As the sum of the hypothenuse (or distance) and half the 
greater leg. (whether difference of latitude, or departure) is to 
the less leg. so is 86, to the sine of the angle opposite the less 
leg. 



94 LOGARITHMLCK 

4. What is the circumference of a circular island, whose 
diameter is 45 rods ? Ans. 141 rods, 1^ yard. 

5. What is the whole distance of space, through which the 
planet Hershel moves, in revolving round the center of the sys- 
tem, supposing its orbit to be a circle, whose diameter is 1,800 
millions miles ? Ans. 11,309,724,000 miles* 

PROBLEM VI. 
To find the diameter of a circle from its circumference. 

RULE. 
Divide the circumference bv 3.14159 

OR 

Multiply the circumference by 113, and divide the product by 
555 : Or multiply the circumference by .31831, and the product 
will be the diameter. 

Ex. 1. If the circumference of the earth be 24913 mile$, 
what is the diameter ? Ans. 7930 miles. 

2. If the periphery of a wheel be 6 feet 6 inches ; what is its 
diameter ? Ans. 2 feet ■£? inches. 

3. If the circumference of the Sun be 2,800,000 miles, what 
is his diameter ? Ans. 891,267 miles. 

4. If the circumference of the Moon be 6850 miles, what is 
her diameter ? Ans. 2 180 miles. 

t 5. If the whole extent of the earth's orbit be 596,902 % 10O 
miles how far are we from the Sun ? Ans. 95,000,O00miles.. 
PROBLEM VII. 
To find the Area of a circle. 
RULE. 
Multiply the square of the diameter by .7854 

OR 

Multiply half the diameter into half the circumference. 
Ex. 1. What is the area of a circle whose diameter is 623 ? 

Ans. 304836 
2/ How many acres are there in a circular island, whose di- 
ameter is 124 rods ? Ans. 75 acres, 76 rods. 
- 3. What is the area of a circle, whose diameter is 7 feet. 

Ans. 38.4846 
,4. How many square ards yare in a circle, whose diameter is 
Si feet?, Ans. 1. 069 

5. What is the area of a circle, whose diameter is 1, and 
whose circumference is 3. 14159 ? Ans. .7854 

If the diameter of a circle is not given, the area may be found 
by multiplying the square of the circumference by .07958. (Sup 
Euc. 8. 1.) 
. Ex. 1. What is the area of a circle, whose circumference is 
1 36 feet ? A&5.- 1 472 feet;. . 



ARITHMETIC*. *s 

2. What is the surface of a circular fish pond, which is 10, 
rods in circumference ? Abb. 7. 95800 rods. 

PROBLEM VIII. 

To find the diameter 9/ the earth, from the known height of a 
distant mountain, whose summit is just visibUb in the horizon, 

RULE. 

•From the square of the height, subtract the height. 

Ex. U The summit of Mount Chimhorazo in South America, 
is about 4 miles above the level of the ocean. If a strait line 
from this touch the surface of the water at the distance of 178£ 
miles ; what is the diameter of the earth ? Aas. 7940 miles. 

2. The White Mountains in New-Hampshire are about 7100 
feet high above the level of Connecticut River ; and a strait line 
from the summit of the mountains will touch the surface of the 
water at the distance of 103^ miles, what is the diameter of the 
earth? Ans. 7940 miles, 

PROBLEM IX. 

To find the greatest distance at which a given object can bo 
seen on the surface of the earth. 

RULE. 

To the product of the height of the object into the diameter 
of the earth, add the square of the height, and extract the square 
root of the sum. 

Ex. 1. If the diameter of the earth be 7940 miles, and 
Mount JEtna 2 miles high, how far can its summit be seen at 
sea ? t Ans. 126 miles. 

2. Suppose the diameter of the earth as in the first example*; 
at what distance may a steeple be seen 9n level ground, allow- 
ing it to be 1 65 feet in height ? Ans. 2 1-j. miles. 

If a man standing on a level plain, has his eye elevated 5* feet 
above the ground ; to what distance can he see the surface of 
the plain ? Ans. 2-J miles. 

4. The top of a ship's mast 132 feet high is just visible in the 
'horizon, to an observer on the deck of another ship 33 feet from 
the surface of the water ; how far are they asunder ? 

Ans. 2 lj. miles. 



•Note 1. See Euclid's Eliments, 36. 3* 

Note. 2. The actual distance at which an object can be seen, is 
increased by the refraction of the rays of light in the air. (See 
End. Nat. Pkil.) But if no allowance be made for this refraction., 
the distance to which a person can see the plane surface of the 
deean, is equal to a tangent to the earth drawn from the ob- 
lervtr's eytf. 



9t LOGABITHMICK 

PROBLEM X. 

Tojind the jfrea of a Triangle. 
RULE. 
Multiply the base of the given triangle into half its perpen- 
dicular height ; or half the base into the whole perpendicular, 
and the product will be the answer. 

Ex. 1. Required the area of a triangle whose base, or longest 
side is 36 inches, and the perpendicular height 16 inches. 

Ans. 36 X 8 =? 289 inches. 
. 2. Required the area of a triangular garden, whose base, or 
longest side is 15.6 rods, and the perpendicular opposite the 
base is 9 rods. .Ans. 70.2 rods. 

PROBLEM XI. 
Tojind the convex eur/ace of a Cylinder* 
*Diffinition. A Cylinder is a round body whose bases are cir- 
cles, like a round column or stick of timber of equal bigness 
from end to end. 

RULE. 
Multiply the length into the circumference of the base. 
Ex. 1 . How many square feet in the superficial contents of a 
cylinder which is 42 feet long, and 15 inches in diameter. 

Ans. 42 X 1.35 X 3.14159 = 164.933 square feet. 
2. Required the convex surface of a cylindrical stick of tim- 
ber, whose axis is 5 feet, and the diameter 7 inches. 

Ans. 1520 inches. 
PROBLEM XII. 
Tojind the solidity of a Cylinder. 

RULE. 
Find the area of the base (by Prob. VII.) which multiply into 
the length, and the product will be the solid contents. 

1. What is the solid contents of a round stick of timber whose 
diameter is 18 inches, and length 20 feet ? 
18 in. = 1.5 ft. 
Xl.5 

2.25 X. 7854= 1.76715 area of base. 
Or 18 inches 20 length. 



18 inches. , Ans. 35*34300 

324 X-7854 « 254*4696 inches, area ef the base. 
20 length in feet. 

• i J' « u ' ■■ 

144)508$-3920(35-343 solid feet, Ans. 
2. What is the solidity of cylinder, whose length is 121, and 
diameter 45.2 ? Ans. 45.3*X-7854X121 = 194156.6 



ARITHMETIC*. 97 

PROBLEM XIII. 

tyfind the solidity of a CotfE. 

Definition. A Con* is a solid yi hose ty*e is a circle, from 
which it decreases gradually to a point in the top, called the 

VERTEX. 

A-line drawn from the vertex, perpendicular to the base, is call- 
ed the height of the cone. 

RULE. 
Multiply jthe area of the base by the height, and £ of the 
product will be the content. 

Ex. i. What |3 the soliclity of a cone, whose height is. 1? 
feet 6 inches, and the diameter of the base 2 feet 6 inches I 

25* X-7854 X 12-5 ~ 3 = 20-453125 feet, Ans. 
2. Required the solidity of a conical monument, that is 9 feet 
high, and the diameter of its base 2£ feet. 

Ans, 14-726250 feet. 
PROBLEM XIV. 
To find the solidity of a Frustrum of a eone. 
Definition. A Frustrum of a cone is what remains after any 
portion of the upper end is cut off, by a plane paralell to the 
base. 

RULE. 
Add together the areas of the two ends, and the square root 
of the product of these areas ; and multiply the sum by £of 
the perpendicular height, and the result will be the solid con*' 
tent. 

OR 

2. Divide the difference of the cubes of the diameters of the 
two ends, by the difference of the dtameters,*and this quotient} 
being multiplied by -7854 and again by \ of the height, will give 
the solidity. 

EXAMPLES. 

1. Required the solidity of a frustrum of a cone, whose altitude, 
or height is 18 feet, the greatest diameter 8 feet, and the least 4 
{pet. By the 1st. Rule. 

89 X -7854 = 50*2656, = area of base. 
4* X -7854 =s 12*5664, = do. of the other erit. 
yM 2*5664 X 50-2656 = 25*1328, = J the pro'd of the 2 areas.. 
Multiply 87*9648 the sum 

by ^ of 18 ~ 6 



Ans. 527*7888 solid inches..' 



By the 2d. Rule. 

.8?— ,4* s 448 -~ (8—4) = 112 X 'P854 % 6 = 

52?-783$ ift. Ai>9., 

13 



98 LOGARITHMICK 

The latter method, i* many cases, will be found preferable tf 
the former in point of expidition. • 

2. .What is the content of the frustrum of a conical blocks 
whose height is 20 inches, and the diameter of its two ends 28 
and 20 inches ? Ans. 9131-5840^ 

The number of gallons or bushels which a vessel will contain 
may be found, by calculating the capacity in inches y and then 
dividing by the number of inches in 1 gallon or bushel ; as by 
the following 

TABLE OF SOLID MEASURE. 



1728 


cubic inches . 


= 1 cubic foot^ 


27 


cubic feet 


= 1 cubic yard, 


4492} 


cubic feet 


= 1 cubic rod, 


32768000 


cubic rods 


= 1 cubic mile, 


282 


cubic inches 


= 1 ale gallon, 


231 


cubic inches 


= 1 wine gallon, 


2150-42 


cubic inches 


= 1 bushel, 



1 cubic foot of pure water weighs 1000 ounces, Avoir- 
dupois, or 62| pounds. 

EXAMPLES. 

1. What is the capacity of a conical cistern, which is 9 feet 
deep, 4 feet in diameter at the bottom, and 3 feet at the top I 

Ans. 87*18 cubic feet X 7-4805* = 652*15 wine gallons. 

2. How many gallons of ale can be put into a vat in the form of 
a conic frustrum, if the larger diameter be 7 feet, the smalle 
diameter 6 feet and the depth 8 feet ? Ans. 1886*5458 gallons. 

3. There is a cistern in a distillery whose altitude is 10 feet 
the greater diameter 14 feet, and the smaller diameter 12 feet, 
required its capacity in hogsheads. 

143— 12* — 14—12 X -7854 X *■*/ X 7-4805 -s- 63 =: 

Ans. 157*918193 hhd. 
PROBLEM XV. 
To find the surface of a Sfihere. 

JDeffimtion. A Sphere, or globe is a round solid- body, in 
the center of which is a point, from which all lines dratorn to> 
the surface are equal. 

RULE. 
Multiply the diameter by the circumference. 
Note. In like manner, the convex surface of any zone or seg- 
ment is found by multiplying its height by the whole circum- 
ference of the sphere. 



* Note. When the capacity is in feet, multiply by 7-4805, be- 
cause 1 J|| = 7*4805 the number of wine gallons in 1 cubic food 
When the ale gallon is required, multiply the feet by 6-1276, 
because ■*} f f = 6-1276 ; but if the capacity be calculated in 
inches divide by the number of cubic inches, in tfce gallon. 



ARITHMETIC^ ♦ 4? 

EXAMPLES. 

1; What is the convex surface of a sphere, whose diameter 
is i inches, and circumference 22 inches ? 

: Ans. 7 X 22 =154 in. 

2 Required the surface of a globe, whose diameter or axis, 
is 24 inches, 24 X 3-14159 X 24 = 1809-5616 inches, Ans. 

3. Considering the earth as a sphere, whose circumference is 
2500$ miles ; how many square miles are there on its surface ? 

An»i 198943750sq. miles. 

4. The axis of a sphere being 42 inches, what is the convey 
superficies of the segment, whose height is 9 inches ? 

Ans. 42 X 3- 14156 X 9 = 1 187-5248 inches. 

5. If the circumference ofthfe sun be 280000a miles, what i5 
the surface?: Ans. 2495547600000 sq. miles. 

PROBLEM XVI. 
To find the solidity of a Sphere* 
RULE. 

1. Multiply the cube of the diameter by *5236. 

OR 

2. Multiply the square of. the diameter bf £ of the circum* 
ference. 

OR' 

3. Multiply th« surface by \ of the the- diameter. 

EXAMPLES. 

1. What is the solidity of a sphere, whose diameter is 1 foot? 

12? X -5236;= 904-7808. inches, Ans. 
; Or 12» X 6-28318 = 904-7808 inches, 
Or 455-38896 X 2 = 904-7808 inches^ 

2. What is the solid content of a sphere 4 feet 6 inches in 
iameter ? Ans. 47.7 1305 00. feet. 

3. Required the number of solid miles contained in the earth, 
Supposing its circumference* to M& 2500D miles:"" 

Ans. 263858149120 miles. 

4^ How many wine gallons will fill a hollow sphere 2 feet 8 
inches in diameter ? 

The capacity is 9.9288 feet X 7:4805,= 1 hhd.l 1.27 gallbns. 

5. How many, gallons of water may be put into a hollow 
sphere that is 4 feet in diameter, and what will be the weight of 
the water ? 



Note. The numbers 3*14159, *7*54, *5236, should be made 
perfectly familliar. The first expresses the ratio of the cir- 
cumference of a circle to the diameter ; the second^ the ratio of 
the area of a circle to the square of the diameter ; and ^e. 
third, the ratio of the solidity of a sphere to the cube of tlflih 
ameter. 

The second is £, and the third is £ of the first. 



t6» LOtJAKITfiMICK 

Amv 905.33832704 gallons, and the weight is 12833.6*544 lfei 

6. If the diameter of the moon be 2180 miles, what is her 
ablidity ? Ans. 5424600000 miles. 

, When the solidity o{ a sphere is given, the diameter may be 
found by dividing the solidity by .5236, and Extracting the cube 
root of the quotient. 

7. What is the diameter of a sphere, whose solidity is 6 $.45 
cubic feet ? W'H'H ~* fi feet Ans. 

8. What must be the diameter of a sphere, to contain 105f 
gallons of wine ? Ans. 3 feet. 

9. Required the diameter of a globe, to contain 16755 pounds 
of water. Ans. 8 feet. 

4 10. How imtny globes that are 3 inches each in diameter, are 
equal to another globe whbse diameter is 12 inches ? 

Ans. 64- 

Note. The solid contents of similar figures are in proporti&n 
to each other, as the cubes of their homologous* sides, or diam- 
eters. Euc. El. 

12. If a cannon ball 6 inches in diameter, weigh 33lb. what 
Will another ball weigh, whose diameter is 3 inches ? 

6 3 = 21^ and 3* = 27, then as 216 : 32 : : 27 : 4 lb. Ans. 

13. If a metalic globe 8 inches in diameter, weigh 72 lb. 
what will be the weight of a globe of the same foetal, whose di- 
ameter shall be 4 inches ? ' Ans. 9 lb. 

. 14. If a globe of silver 3 inches in diameter, be worth 8*50^ 
fibw nferoy such globes will be equal in value to £9600 JT 

Ans. \4. 



ANNUITIES, OR PENSIONS.. 

An Annuity, is a ium of money payable every year, for a 
certain number of years, or forever. 

When the debtor keeps the annuity in his own hands beyond 
the time of payment, it is said to be in Arrears. 

The sum of all the annuities for the time they have been for> 
born together with the interest due upon each, is called the 
Amount. 

If an aftnuity be bought off? or paid all at once, at the begip. 
tring of the first year, the price, which ought to be given for it, 
is called the Present Worth. 

To find the amount of an AnkVjty at Stm?le I^ter^St. 



ARITHMETIC*. itfT 

RULE. 

1, Find the sum of the natural series of numbers, 1, 2', S, &ti 
up to the giren number of years wanting tmc. 

2. Multiply this sum by one year's interest of the annuity, and 
the product will be the whole interest due upon the annuity. 

3* To this product add the product of the annuity multiplied 
Into the time, and the sum will be the amount sought. 

EXAMPLES. 

1. What is the amount of an annuity of £50 for 7 years, aT- 
Ibwing simple interest at 5 pr. cent ? 

1 +2 +34-4 4.54. 6 = 21=3X7 
£2 10s. = 1 years interest of 50f. 
3 

7 lit 

7 

52 10 
350 0=3 £"50X7 

£402 10 = amount required. 

2. If a pension of g600 pr. annum be forborn S years, what 
% will it amount to, allowing 6 pr. cent simple interest ? 

Ans. 83360 
3* If a salary of 8750 annually, remain unpaid for 4 years, 
how much mu£t be paid at the end of said term, allowing 4£ pr. 
cent simple interest. Ans. $3202 50 

Tojind the jiretent worth of an Annuity at Simfile Intercat. 

RULE. 
Find the present worth of each year by itself, discounting from 
the time it becomes due, and the sum of all these will be the 
present worth required. 

EXAMPLES. 

^ 1. What is the present worth of an annuity of glo6, to con- 
tinue 5 years, at 6 pr. cent pr. annum simple interest ? 
1 06"! T94.&39 6 = present worth for 1 st year. 

112 j 89.2S57 =2 « 2d year. 

128 U-: 100 : * 100 : ^ 84.7457 «s " 3d year. 

124 180.6451= ** 4th year. 

J30j \J6.92S0 a: « *th year. 

425.9391 a= 35425 93 CtS. 9m. T ^ 
present worth of the annuity required. 

2. What is the present wtfrth of $400 pr. annum, to continue 
4 years at 6 pr. cent ? Ans. 81396.065^3 



j 



tar LOGARITHMICK 

3. What is the present worth of an annuity, or pension of 
jf 500, to continue 4 years, at 5 pr. cent pr. annum, simple inter- 
est? Ans. f 1782 3s. 8|d. 

To find thh Amount of an Annuity at Compound interest. 
RULE. 

1. Make 1 the first term of a geometrical progression, and 
the amount of £1. or SI for one year, at the given rate pr. cent, 
the ratio. 

2. Carry the series to as many terms as the number of years* 
and find its sum. v 

3. Multiply the sum thus found by the given annuity, the 
product will be the amount sought. 

EXAMPLES. 

i. If a salary of g600 be forborn (or remain unpaid) 7 years ; 
what will it amount to at 6 pr. cent pr. annum, compound inter- 
est ? 1 + 1 .060000 J- 1.123600 +1.19101 16+1.262476 + 1.338225 
►1-1.418519+1.503630= 8.897466 = sum of the series.* 
Multiplied by 600 
gives 5338.4796 = g5338 47 cts. 9/ ff m* the amount 
sought. 

Or, By Table III. 
Multiply the Tabular number under the rate, anjj opposite to 
the time, by the annuity, and the product will be the amount 
sought. 

2. If a pension of g 175 pr. annum, be forborn 20 years, at 6 
pr. cent compound interest ; what is the amount ? 
Tabular number = 36.785590 

175 = Annuity. 



An> 6437.478250 = 86437.47 C. 8 J m. 



3. Suppose g 50 pr. annum, with compound interest at 5£ pr. 
cent.be 10 years in arrears ; required the amount. 

Tabular number = 12-875354 X 50 = g 643-76c.7m. Ans. 

4. What will a pension of £ 120 pr. annum, amount to in 3 
years, at 5 pr. cent, compound interest ? Ans. £378 6s. 

5. The salary of the President of the United States, is g 25000, 
supposing the whole be in arrears during the period of his elec- 

*The sum of the series thus found, is the amount of £l 9 or 
gl annuity, for the given time, which may be found in Table 
I Hi ready calculated. (The method of constructing these To* 
dies vnll be shown hereafter.) y 

Hence, either the amount, or present w^orth of annuities; 
continuing for a term not exceeding 40 years, may readily be* 
found by the Tables for that purpose. ' 



ARITHMETIC**. 1Q3 

tSJm, or 4 years ; what would then be the amount of his salary, 
allowing 6 pr. cent, compound interest ? Ans. g 109365,40 

To find the present worth of Annuities at Compound Inter est '• 
RULE. 

1. Divide the annuity by the ratio, or the amount of g l,er 
£\ for one year, and the quotient will be the present worth of 
the first year's annuity. 

2. Divide the annuity by the square of the ratio, and the quo- 
tient will be the present worth of the annuity for the second 
year. 

3. Find, in like manner, the present worth of each year by 
itself, and the sum of all these will be the present worth of the 
annuity sought. 

EXAMPLES. 

1. What is the present worth of an annuity of g 40 to con- 
tinue 5 years, discounting at 5 pr. cent. pr. annum, compound 
interest ? 

Ratio) 1 :-: 1-05)40* 00000(38095= present worth 1st. year. 

Ratio)*— 1- 1025)40- 00000(36-281 = « 2d. year. 

Ratio)*-- 1-57525)40' 00000(34-556 = « 3d. year. 

Ratio)* — 1-2 15506)40- 00000(32 899= « 4th. year. 

Ratio)* = 1-2762 18)40. 00000(31-342 = " 5th.year. 

173173= g 173 17c. 3m. whole 
present worth of the annuity required. 

Or, By Table IV. 
Multiply the tabular number under the rate and opposite to) 
the time, by the annuity, and the product will be the present 
worth required. 

2. What is the present worth of an annuity of g 50 to con- 
tinue 5 years, at 6 pr. cent, pr. annum, compound interest I 

Tabular number = 4-21236 X 50 = Ans. g 210 61 8 

3. If the pension of an officer, serving in the Revolutionary 
War, be 20 dollars a month, or 240 dollars annually ; what is 
-its present worth, allowing a discount of 6 pr. cent, pr. annum, 

compound interest, supposing it to continue 10 years ? 

Ans. g 1766 41c. 9^m. 
To find thefire8ent worth of a Freehold Estate, or an 
Annuity to continue forever, at Compound Interest. 

RULE: 
As the ratepr. cent, is to g 100, so is the yearly income u 
the value required. 

I. What is the worth of a freehold estate of g 40 pr. annui* 
allowing 5 pr. cent, to the purchaser ? 

As 5 > 100 : j 40 : g 800 Ans. 



1*M LOGARITHM1CK 

• * 

a. What >• a freehold estate of £7§ a year worth, allowing 
the buyer 6 pr. sent, compound interest for bis money ? 

Ans. £ 1250 

3. An estate brings in yearly 8 79*2a what would it sell for, al- 
lowing the purchaser 4| pr. cent, compound interest for his 
money ? Ans. 81760. 

To find the fir ceent worth of an Annuity, or Freehold E*ta& 
in Reversion at compound interest. 

RULE. 

Find the present value of the estate (by the foregoing rule) 
as though it were to be entered on immediately, and divide said 
value by that power of the ratio denoted by the time of rever- 
sion, and the quotient will be the present werth of the Estate 
in Reversion. 

EXAMPLES. 

1 . The reversion of a freehold estate of £ 79.4s. pr. annum, to 
commence 7 years hence, is to be sold ; what is it worth in 
ready money, allowing the purchaser 4} pr. cent, for his money? 

As 4-5 : 100 * .: 79-2 : 1760 » present worth, if entered on 
immediately. 

And 1045> 7 = 1- 360862) 1760-0d0( 1203*297 = £ 1293. 5s. 
1 1 Jd. = present worth of 17601. for 7 years, or the whole pres* 
ent worth. 

Or, By Table IV. 

Find the present worth af the annuity, or rent, for the time of 
reversion, which subtract from the value of the immediate pos- 
session, and the remainder will be the value of the estate in 
reversion. 

2. What if the present worth of a freehold estate of g 40 pr. 
annum, to commenpe 7 years hence allowing the purchaser 5 
pr. cent ? 

Tabular number = 5-78-637 

40 at annnky, or rant. . 

231*45480 =c present worth of rent. 
& : ,100 : : 40 ; 8000000 = value of immediate possession. 

5685452 as g 568- 54c. 5|m. An*. 

3. Whkh is the most valuable, an income of £200 pr. amium 
Br 15 years, or the reversion of an equal income forever after* 
ward, computing at the rate of 5 pr. cent, pr. annum, compound 
ihterest ? 

Ans. The first term of 15 years is better than the reversion 
fflreyer afterward, by 17/. 18s.7*d» 



ARITHMETICS 105 



VULGAR FRACTIONS. 

The learner will find it convenient, and indeed necessary, as 
lie progresses in the more intricate parts of Arithmetick, to be 
thoroughly'acquainted with Vulgar fractions. They were brief- 
ly introduced immediately after Simple Proportion (page 56) as 
a preliminary to the subject of Decimal Arithmitick ; but we 
shall here consider, more extensively, the intimate relation be- 
tween Vulgar, and Decimal Fractions, together with their rela- 
tive and important use in Arithmetick in general. 

Vulgar Fractions are either proper, improper, simple, 
compound, or mixed. 

1. A Simple , or proper fraction is one, whose numerator is 
less than the denominator ; as f ,£,£, Sec. 

fc. An Improper fraction is one, whose numerator exceeds the 
denominator ; as f ,}|, Sec. 

4. A Mixed JVumder is composed of a whole number and 
a fraction ; as 81,25^, &c, 

Note. Any whole number may be expressed like a fraction, 
by writing. \ under the givem number for a denominator j as 
8= f and 12 = y, &c. 

3. A Compound fraction is the fraction of a fraction, coupletf 
by the word of as ; | of J of i of y, Sec. 

5. The Common Measure of two or more numbers is that 
, number, which will divide each of them without a remainder. 

Thus, 4 is the common measure. of 12, 16, and 20 ; and the 
greatest number^ that will do this, is called the Greatest Ctrnmon 
Measure. 

6. A number, which can be measured by two or more num- 
bers, is called the Common Multiple ; and if it be the least 
number, which can be so measured, is called. their Least Com- 
mon Multiple : thus, 24, 36, 48 and 60 are each a common muly 
tiple of 3, 4, and 6 ; but their least common multiple is 12. 

To find the least Common Multiple of two or more numbers. 

RULE. 

1. Divide by any number that will divide two or more of the 
given numbers without a remainder, and set the quotient| to- 
gether with the undivided numbers, in a line beneath. 

2. Divide the second lines as before, and so on till there are 
no two numbers that can be divided ; then, the continued pro- 
duct of the divisors and quotients, will give the multiple re- 
quired. 

14 



106 LOGARITHMICK 

EXAMPLES. 

l. What is the least common multiple of 6, 8, 10, and 12. 

2)6 8 10 12 



Thus. 



3)3 4 5 6 

2)1 4 5 2 
12 5 1 

The product of the divisors = 12 and 12X2 X 5 120 Ans. 

2. What is the least common multiple of 4 and 63 ? 

Ans. 12 

3. What is the least common multiple of 3,4,8, and 12 ? 

Ans. 24 

4. What is the least common multiple of 4,5,6, and 10 ? 

Ans. 60 

5. What is the least number that can be divided by the 9 
digits, separately, without a remainder ? Ans. 2520 



REDUCTION OF VULGAR FRACTIONS: 

Is the bringing them out of one form into another, in order to 
prepare them for the operations of Addition, Subtraction, fee. 

CASE I. 
To abbreviate y or reduce fractions to their lowest terms* 
RULE. 

1. Divide the terms of the given fraction by any number, that 
will divide them without a remainder, and these quotients again 
in the same manner ; and so on till it appears, that there is no 
number greater than I, which will divide them again, and the 
fraction will be in its lowest terms. 

OR 

2. Divide both the terms of the fraction by their greatest 
common measure, and the quotients will be the lowest terras of 
the fraction required. 

EXAMPLES. 

1. Reduce ift to its lowest terms. 
(2) (2) (3) (2) (2) 
*H = t¥* ^U = H = A «■? the answer.— (see page 56) 



ARITHMETIC*: JOT 

Or thus; 144)240(1 
144 

96)144(1 
96 

48)96(2 
48 

Therefore 48 is the greatest common measure, and 48)^|f = J 
the same as before. 

2. Reduce ffff to its lowest terms. Ans. f 

3. Reduce Jf $$ to its lowest terms. Ans. £ 

4. Reduce §£f to its least terms. Ans. -^ 

5. Reduce £$f to its lowest terms. Ans. ££ 

CASE II. 
To reduce a Mixed Number to its equivalent improper fraction!. 

RULE. 
Multiply the whole number by the denominator of the frac- 
tion, and add the numerator to the product, then that sum writ- 
ten above the denominator will form the fraction required. 

EXAMPLES. 

1. Reduce 25f to its equivalent improper fraction. 

25X8 + 3 = 193, then the fraction will become « | c 

2. Reduce 27f to its equivalent improper fraction. 

Ans. *$* 

3. Reduce 45 J to its equivalent improper fraction. 

Ans. 3|7 

4. Reduce 100$f to its equivalent improper fraction. 

Ans. *f|9 

5. Reduce 15|$ to its equivalent improper fraction. 

Ans. « T » 
CASE- III. • 
To find the value of an improper fraction. 

RULE. 
Divide the numerator by the denominator, and the quotient 
will be the whole or mixed number sought. 

EXAMPLES. 

1. Find the value of 3 |f 8 21)38.48(183^ Ans. 

2. Find the value of 'A 8 " Ans. 9. 

3. Find the value of x i$ s Ans. 56 U 

4. Reduce * r y to its equivalent whoje, or mixed number. 

Ans. 84 & 

5. Reduce 4 |^ f to its equivalent whole, or mixed number. 

Ans. 173 A 
CASE IV. 
To reduce a whole number to an equivalent /ractuta* hvovnfc a 
given denominator. 



10S LOGARITHMICK 

RULE. 

Multiply the whole number by the given denominator, and 
place the product over the said denominator, and it will form 
the fraction required. 

EXAMPLES. 

1. Reduce 8 to a fraction, whose denominator shall be 9. 

8X9 = 72 ; and the fraction will become V Ans « 

2. Reduce 13 to a fraction, whose denominator shall be 12. 

Ans. Vt 

3. Reduce 100 to a fraction, whose denominator shall be 79. 

Ans. 7 ^° 
CASE V. 
To reduce a compound fraction to an equivalent simple one. 
RULE. 

1 . Reduce all whole and mixed numbers to their equivalent 
fractions. 

2. Multiply all the numerators together for a new numerator, 
and all the denominators together for the denominator, and they 
will form the fraction required. 

EXAMPLES. 

1. Reduce 4 of £ of £ of f- M to a simple fraction. 

2X3X4X8= 192 

■ ■ ■■ m — = j-f the answer. 

3 X 4X5 X 11 =660 

2 . Reduce £ of 4 of £ of ^ to a simple fraction. Ans. ^ 

3. Reduce i of £ of 10 to a simple fraction. Ans. 4 J 

4. Reduce 4 .oi f of & of 2 If to a simple fraction. 

Ans.2ifi* 
CASE VI. 
To reduce fractions of different denominations to equivalent 
fractions^ having a common denominator. 

RULE. 
1 Reduce all fractions to simple terms. 

2. Multiply each numerator into all the denominators, except 
its own, for a new numerator ; and all the denominators togeth- 
er, for a common denominator, which written under the several 
numerators, will give the fractions required. 

EXAMPLES. 

,1. Reduce \% s and $ to equivalent fractions, having a com- 
mon denominator. 

1 X 5 X 7 = 35 the new numerator for i 
3X2X7 = 42 ,: « for f 

4X2 X 5 = 40 « for 4 

2X 5 X 7 = 70 the common denominator. 
Therefore the new equivalent fractions are ^f, 4o» an( ^ H 
2. Reduce J, T V and }£ t0 a common denominator. 



ARITHMETICS tO» 

3. Reduce |, J, and 1 2£ to a common denomlnatoK 

4. Reduce |, f of J, 5t> and^ to a common denominator. 

5. Reduce |,| f £, 7£* and ^ to a Common denominator. 

Ans. tfrVtHt'Wtf'tfk 

CASE VII. 
To reduce the fraction of one denomination to tjie fraction of 
another, retaining the same value-. 

RULE. 

1 . Reduce the given fraction to such a compound one* as will 
express the value of the given fraction, by comparing it with ail 
the denominations between it and that denomination to which it 
is to be reduced. 

2. Reduce the compound fraction) thus made, tp a simple 
one. (See Case V.) 

examples. , 

1. Reduce $ of a penny to the fraction of a pound. 
By comparing it, it becomes £ of T V of T J of a pound. 

3X1X1 
- i v»- = ytfVy the answer. 

8 X 12X20 

2. Reduce T ^ of a pound to the fraction of a penny* 
Make a compound fraction of it thus ; 

TtfW of V of Y = tW* = * the Ans. 

3. Reduce f of a shilling to the fraction of a pound* 

Ans* J* 

4. Reduce f of a farthing to the fraction of a pound* 

Ans. r J in 
5. Reduce f of a pound avoirdupois to the fraction of a cwt. 

* ■ \ Ans* - ¥ f T 

6. Reduce T7 \ T of a hhd. of wine to the fraction of a pint. 

Ans. ^ 

7. Reduce S\ furlongs to the fraction of a mile* Ans. |j- 

8. Reduce T ^ 7 of a week to the fraction of an hour* Ans. $ 

9. Reduce 7s. 6d. to the fraction of a pound. Ans. $ 

10. Reduce 5}d. to the fraction of a shilling. Ans. }| 



ADDITION OF VULGAR FRACTIONS. 

RULE* 
Reduce compound fractions to single dnes ; mitced numb* 
to improper fractions ; fractions of different integers to thot 
of the same ; and all of them to a common denominator : tht 
15 



J 



110 LOGARITHMS*. 

the sum of the numerators written over the common ^ieflominar 
tor, will be the sum of the fractions required. 

BKAMPI&&. 

Add 3fr, $, | of }, and 7 together. 
First 3f « y,4of£ = fti-r ** J 

Then the factions are y , J , T ^, and *. Xherefore, 

29X8XlOXl=a 93*20 

7 X8 X 10 X 1 «= 560 

7 X 8 X 8 X I === 448 

7 X8 X 8X10 =3=4480 



ir 



7808 

— «= 12} the answer 
8X8X10X1= 640 

2. Add £, ^, and £ of 5* together. Atis. ^> n 

3. Add 12 J, 3§, and 4 J together. Ans. 20 ji 

4. Add $1. Js. and yj of a penny together. 

Ans. 3s* id. i|fqrs. 

5. What is. the sum of $ of 15/. 3f/. £of f of f«f a pound, 
and | of ^ of a shilling ? Ans. £i 17s. 5|d. 

6. Add | of a mile, § of a yard, and J of a foot together. 

Ans. 120 rods, 2 ft 9 in. 

7. Add 4 of a ton, and ^ °f cwt « together. 

Ans. 12«wt. 1 qr.8lb. 12^.«b. 

8. Add a of a week, { of a day) and $ of atn hour together. 

Ans* 2d* 14h. #0m. 



SUBTRACTION OF VULGAR FRACTIONS. 

RULE. 
Prepare the fractions as in addition, and the difference of the 
numerators, written above the common denominator, will gige 
the difference of the fractions required. 

EXAMPLES. 

1. Fromf take f of -f. 

i of ^ = ^r ; then the fractions are ^ r and f, therefore 
2 X 3 a 6 ("and 42 — 6= 36 

21 X 2 = 42 J Ans. f f = $ 

and 21 X 3 = 63 the common jdenomihator. £ 

2. From f& take ^. , Ans. U& 

3. From 14^ take ^ of 19. Ans. 1 T 7 T 

4. From \ L take £ *. Ans. 9s. 3d. 

5. From -f oz. take -J of a'pwt Ans. 1 1 pwt. 3gr. 



ARITHMETICS in 

*. From 3$ weeks, take \ of a day, «nd i of * of $ of an hour. 

Ans. 3vr.4d. I2h. 19m. 17^ sec. 

7. The sum of three number* is 56} ; thejirst number is 12-J 
atid the second 2l T ^ ; required the Mzrof. Ans. 22 T V T 

8. What number added to 1 1 4 will make 36$f % ? 

Ans. 24ft. 



MULTIPLICATION OF VULGAR FRACTIONS, 

Reduce compound fractions to single ones, and mixed num- 
bers to improper fractions ; then multiply all the numerators to- 
gether for a new numerator, and all the denominators together 
for the denominator of the product required. 

EXAMPLES. 

1. Multiply i of 7 by I - Ans> 11 

2. Multiply \ by \ Ans. «J.; 

3. Multiply 7i by 9J Ans. 69f 

4. Multiply I of i by £ of 3£ Ans,-^ 

5. Multiply 4i \ of i and 18 f continually together. 

Ans.ft^ ? 



DiyfSION OF VULGAR FRACTIONS. 

RULE. 

Prepare the fractions as in .multiplication ; then invert the 
djyisor, and proceed exactly as in multiplication 1 the product 
Twill be the quotient required. 

examples. 

1 . Divide f of 9 by f of 7\ 

2 X 9 
j of f = — .. ^ V **& $ of H—Hl therefore, 
5X1 
18X16 

V -5-if = " *= 4*1 Wy the quotient required., 

5X45 

2. Divide f by §. Ans. I/ r 
£. Divide \ by 4. Ans. ^V 

- - - - Ans. 2 ? V 

Airs. )8*£ 



4. Divide 4$ by f of 4, 

5. Divide 7 by >. 



112 LOGARITHMICK 

6. Divide | of 1 9 by | of |. Ans. 7f 

7. Divide i of f by | of }. Ans. f 

8. What number multiplied by f , will make 1 1^ ? 

Ans. 26$f- 



SIMPLE PROPORTION IN VULGAR FRAC- 
TJONS. 

RULE. 

1. Prepare the fraction as. before ; then state the question, 
agreeably to the rules in Simple Proportion of whole numbers. 

2. Consider whether the proportion be Direct or Inverse ; if 
direct, then invert the Jirst term of the proportion ; but if. the 
proportion be inverse, invert the third term. 

3. Then multiply all the three terms continually together, 
and the product will be the answer. 

examples. 
1 . If | of a yard cost f of a pound, what will j| of a yard cost ? 

Thus, | ; £ : : || : jfj = 12s. Id. 2^ qrs. Ans. 
2; If | of a yard cost ffof a pound, what will 91 yards cost ? 

Ans. £4 10s. 2d. 2|qrs. 

3. If Sd.buy j of a pound of sugar, how much will 10£d. buy T 

Ans. lib. 

4. At 7s. Od. If | qrs. pr. bushel, what will be the value of 15 
bushels ? £5 5s. ?d. 0£ qr. 

5. If | of a ship be worth $ of her cargo valued at 8000/. what, 
is the whole ship and cargo worth ? Ans. £ 10031 Hs. ll T \-d. 

6. A- and B. own a ship and cargo worth 16000/. A owns 
J of the cargo and ^ of the ship ; but by accident at sea, they 
lose \ of the cargo, which is 214f/ t less than B's. share in the 
ship, required the values of the ship and cargo, and each one's 
respective share in the same. 

Ans. j£*4000 value of ship, and 12000/. do. of cargo. 
A's share of cargo = 3937-^. C B's share of cargo = 6562*/. 
A's do. of ship = 22854/. C B 's do. of ship = 17l4f/; 

A*s do. of the whole £622&ft B's do. of the whole £8276}J 



ARITHMETICS tis 

REDUCTION OF DECIMAL FRACTIONS. 

CASE I. 

To reduce numbers of different denominations to their equiva- 
lent value. 

RULE. 

Bring the givtn denominations 10 a vulgar fraction, and reduce 
said fraction to its equivalent decimal value. (See Case lily 
page 57, also Rule 2, page 64.) 

EXAMPLES. 

1. Reduce 10s. 6d. 2 qrs. to the decimal of a pound. 

1/. X 20 X 12 X 4 « 960 and 10s X 12 X 6 X 4 -h 2 = 506, 
therefore, 506 — 960 = .527085 Ans. 

2. Reduce 13s. 5£d. to the decimal of a pound. Ans. .6729 
t 3. Reduce 3 qrs. 2 na. to the decimal of a yard. Ans. t 875 

4. Reduce 17 yds. 1 ft. 6 in. to the decimal of a mile. 

Ans. .00994318 

5. Reduce 10 weeks, 2 days, to the decimal of a year. 

Ans. .1972J602, &c. 
CASE IL 
To .find the value of a decimal in the known parts of the integer. 

RULE, 
1. Multiply the decimal byj.he number of parts in the next 
less denomination, and cut off so many places for a remainder, 
on the right, as there are places in the given decimal. 

% Multiply the remainder by the next inferiour denomina- 
tion, and cut off a remainder as before ; and so on through all 
the parts of the integer, and the several denominations stand* 
ing on the left, make the answer. 

EXAMPLES. 

1. What is the value of .37623 of a pound I 

20 



7.42460 
12 




1.18080 Ans. 7s. 6d. 1 qr. 

2. Wl,at is the value of .83229 1 6 of a pound ? Ans. 1 6a. 7-J6\ 

3. What is the value of .625 of a shilling ? Ans. 7£ck 

4. What is the value of .76442 of a pound Troy? 

Ans. 9 oz. 3 pwt, 11 gr. 
£ Find the value of .875 of a yard. Ans. 3 qr. 2 na. 



ll> LOGARITHMIC* 

6.J Find the value of .61 of a ton of wine* 

Ans. 2 hhds. 27 gals. 2 qts. 1 pt. 
7. What is the value of .8469 of a degree I 

Ans. 58 m. 6 fur. 35 po. ft. II in. 
i* JVhat is the value of .569 of a year ? 

Ans. 207 d. 16 h. 26 m. 24 sec. 
CASE IV, 
To find the decimal of any number of 'shillings^ pence , and far- 
things , by jns/iection. 

RULE. 

1. Wrjte half the greatest even number of shillings for the- 
first decimal figure. 

2. Let the farthings in the given penee and farthings, possess, 
the second and third places ; observing to increase the second 
place by 5, if the shilings be odd; and the third place by 1, whea 
the farthings exceed 12, and by 2, when they exceed 36. 

EXAMPLES. 

X. find the decimal expression of 9s. 7±d. by inspection- 
.4 = f 8s. 

.05 = for the odd shilling 
SO = the farthings in 7\jL 
1 for excess of 12: 

£\4Sl ^ decimal required. 

2. What is the decimal value of 17s. 8|d.? Am. •/-.8B$ 

3. What is the decimal expression of 7s. 9Jd. ? Ans. J.S91 

4. Find the decimal value of 151. 3s. 9*d. Ans. £15*19 

CASE V. 
To find the value of any decimal of a pound by inspection* 

RULE. 

1. Double the first figure, or place of tenths in the decimal, 
for so much of the answer in shillings, increasing the sum by 1, 
if the second figure be 5, or more than 5* 

2. After the 5 is deducted, call ,the remaining figures in the 
second and third places, so many farthings, for the remainder of 
the answer, abating I , if they exceed 12, and 2, if they exceed 36.- 

Note. When the decimal has but 2 figures* if any thing re* 
main after the shillings are deducted, annex cyphers on th* 
right. 

EXAMPLES. 

i. Find the value of £.876, by inspection. 
.876 

16s. = double of 8. 

1 5 for the 5 in the second place which* 
£is to be deducted out of 7. 
And 6 f= 26 farthings remain to be added~ 
Deduct i for the excess of 12: 
Ans. 17s. 6 £d. 



ARITHMETICS. 115 

2.. Find the value of/\679, by inspection. Ans. 1 3s. 7d. 
S. Find the value of/\842i by inspection. Ans. 16s. lOd, 

4. Find the value ofjf.790 by inspection. Ans. 15s. 9£d. 

5. Find the value of £.097 by inspection. Ans. Is. Hid. 



SIMPLE PROPORTION BY DECIMALS, 

RULE. 
Seduce fractions to decimals, and state the question as in 
whole numbers ; multiply the second and third terms together, 
*nd divide by the first, and the quotient will be the answer. 

EXAMPLES. 

1. If | of a yard tost -f of a pound, what will 9-1 yard cost ? 

f = .375 yds. 

£ = .41. 

9| = 9.625 yds. 
"Therefore .375 yds. : 41. : : 9.625 yds. : £10.2666 ; or 10L 5s. Sd. 
3 qrs. 

2. If j. of a yard cost T 7 ¥ of a pound, what will T ^- of an English 
*ll cost I £ of a yard = £ of f of f =^| of an ell. 

Then f-f ell : T y. : ; T " r ell. : 44L == 9s. 8d. 2 qrs. Ans. or, ' 

« = 48 f 

7 7 j = .5833 -J Then .48 : .5833 : : .4 ; .486 = 9s. 8d. £rs. 

3. At 7£d. pr. lb. what will be the price of an cwt. of sugar ?' 

7-5 X 1 12 = 840 = 3/. 10s. Ans. 

4. What is the value of 3% cwt. of coffee at 23j cts. pr. lb. \ 

Ans. g98 70c. 

5. What is the value of 2 qrs. 1 na. of velvet at 19s. 8f d. pr. 
English ell ? Ans. 8s. lOd. Iqr. T \ 

6. If \ of a yard of satin cost 7s. 3d. how many yards can 1 buy 
Jor 13/. 15s. 6d. ? Ans. 28f yds. 

7. What is the value of £ of a tun of wine, when £ of a gallon 
x;osts I of a pound ? » Ans. £\40 

8. At li/« P r « c ^t. what does o-Jlb- come to? Ans. iO$d. 
S. What is the tax upon 745/. 14s. 8d. at 3s. 6d. on the pound ? 

Ans. £130 10s. 3}qrs. 

10. A person shares f in a certain prize, and sells f of it for 
171/. what was the whole amount of the prize ? Ans. £3$Q 

1 1. If, when the days are 13f hours long, a traveller perform 
his journey in 35£days ; in how many dayswiH he perform the 
same journey, when the days are 1 l-j\ hours long ? 

Anfc 40A.?4 <h*vs. 



116 LOGARITHMICK 

12. A regiment of soldiers, consisting ©f 976 men, are tojbe 
new clothed, each coat to contain 2J yds. pf cloth, that is If yd, 
wide, and to be lined with shalloon, } yd. wide ; how many yards 
of shalloon will line them ? Ans. 4531 yds- 1 qr. 2^ na* 



SIMPLE INTEREST BY DECIMALS: 

A TABLE OP RATIOS. 

RATE PER CENT. | RATIO, | RATE PER CENT. f RATI<£ | 
4 I 



3 t -03 \ Si 

.04 | 6 

4± t -045 I 6i 

5 I -05 I 7 



06 j 
065 I 

or I 



Ratio is the simple interest of £ J, or g I for one year at the . 
rate per cent, agreed on. 

RULE 

Multiply the principal, Ratio and Time continually together, 
and the last product will be the interest required. 

EXAMPLES. 

1 . Required the interest of 537 dolls, 58 cts. for 4 years artd 6 
months, at 5 per cent. pr. annum, simple interest. 
55537.58 principal 
.05 ratio. 

26.8790 interest for 1 year. 
4-5 multiply by the time* 

1543950 
1075160 

8 1*0.95550 Anjs. g 120 95 cts. 5 m. &c. 



2. WJiat is the interest on 8268 17 cts. for 3 years and 9«, 
months, at 44 pr. cent, simple interest ? 

Ans. 268.17 X .045 X 3.75 = 45.2536875=845 25 cts. 3 m. - 

3. What is the interest of 1181. 9s. for 1 year and 6 month 
at 6 pr. cent pr. annum ? Ans. 101. 13s. 2d. 2 qrs. 

4 Required the amount of 648 dolls. 50 cts, lor 12.75 years 
at 5i pr. cent, simple interest. Ans. &l 103 26 cts. -f- 

5. What is the amount of 691. 8s. for 3 years, 1-J months, at 
£ pr. cent, simple interest ? Ans. £h 1 6s. 6cl. 3 qrs 



ARITHMETICS* ,%V 

CASE II. 
Tkc amount 9 time, and ratio giv<?n, to find the principal. 

ROLE. 
Multiply the ratio by the time, add unity to the product for a 
divisor, by which divide the amount, and the quotient will be 
the principal. 

EXAMPLES. 

1. What principal will amount to 264.3125, in 5 years, at 5 
pr. cent pr. annum ? 

.05 X 5 ± 1 = 1.25) 264.3125 (211.45 Ans. 

2. What principal will amount to 2658.53550, in 44 years, at 
S pr. cent. I . Ans. £537.58 cts. 

3i What principal will amount to 8313.423687$, in 3 years 
and 9 months, at 4$ pr. cent? Ans. 8268 17 cts. 

4. What principal will amount to £956 10s. 4.125d.in 8 years 
and 9 months, at 5§ pr. cent ? Ans. £645 15s. 

CASE III. 

The amount, principal, and time given, to find the ratio. 
RULE. 

Subtract the principal from the amount, divide the remainder 
by the product of the time and principal, and the quotient will 
be the ratio. 

EXAMPLES. 

i. At what rate pr. cent, will 8950 75 cts. amount to 31235. 
9750, in 5 years ? 

From the amount = 1235.9750 
Take the principal = 950.75 



950.75 X 5 = 4753.75 )285.2250(.06 =* 6 pr.cent. Ans. 
285.2250 



2. At wLat rate pr. cent will 87 1 5.45 cts. amount to 894.54940, 
in 2 years and 5 months ? Ans. 6 pr cent. 

3. What rate pr.ct. will 8268. 17 cts. amount to 83 1 3.4236875, 
.in 3£ years ? Ans. 4-J pr. cent. 

4. At what rate per cent will £319 5s. amount to £ 62.253750 
in 3 years and 3 months ? Ans. 6 per cent. 

CASE IV. 
The amount, principal, and rate per cent given, to fipd 
the time. 

RULE 
Subtract the principal from the amount, divide the remainder 
by the product of the ratio and principal 5 and the quotient will 
be the time. 

EXAMPLES. 

1. In what time will 8537 58 cts. amount to 8658.53550, at 5 
pr. cent pr. annum ? 
16 



US JUOGARITHMICK r ^ 

From the amount 2653.55550 
Subtract the principal 537.58 

537.58 X ,05 26.8790) 1 20.95550(4^ year* answer, 
120.95550 



2. In what time will 8268 .17 cents amount to 8313.4236875, 
*t 44 per cent per annum ? Ans. 3 years 9 months. 

3 In what time will 8950 75 cents amount to 81235.9750, at 
6 percent per annum? Ans. 5 years. 

4. In what time will £ 319 5s. amount to £381.503750, at 6 per 
cent ? Ans. 3£ years. 

TO CALCULATE INTEREST FOR DAYS. 
, RULE. 
Multiply the principal by the given number of days, and thai 
product by the ratio ; divide the last producjt by 365, and the 
quotient will be the interest required. 

EXAMPLES. 

1. What is the interest of 1781. 15s. for 87 days, at 6 per cent 
per annum ? 178.75 X .87 X .06 = 836*5500 -~- 365. = 2.2891 
= 21. 5s. 9d. 1 qr. Ans. 

As the process of division is generally more tedious than that 
Of multiplication, it is often convenient to substitute one for the 
other. 

Now if we substitute, in the place of 365, the reciprocal of 
that quality ; that is, the quotient arising from dividing 1 by 
365, we shall have a number which multiplied mto the principal, 
will give the same result, as dividing by its corellative quanti- 
ty ; Thus, 1-s- 365 = '00274, therefore to multiply a given, 
number by '00274,* is equivalent to dividing the same number 
by 365. 

2. What is the interest of 8 100, for 75 days, at 6pr. cent pr. 
annum ? 100 X 75 X -06 X -00274 =81 23c. 3m. Ans. 

3; What is the interest of 8 148 50 cts. for 96 days, at 5 pr. 
cent pr. annum \ Ans. 8 1 95c. 3m. 

4. What is the interest of 8 3 1 2 for 25 days, at 7\ pr. cent ? 

Ans. 8 1 60c. 3m. 

5. What will £V5 amount to in 256 days, at 4^ pr. cent pr. 
annum? Ans. £5 12s li<L + 

%!..* Note. The cyphers on the left may be always neglected, by 
observing to point off 5 figures of the product, on the right for 
decimal parts. 



ARITHMETICS 



U'9 



i 



<p-6~e^ ~ e^^^~^*-&-*-*>- * fy>--** »■ » ' e "" b' " *' w»« " » 




TABLE I. 



Shewing {he A- 
5 mount of 1 Dol : 
J lap, op 1 Pound for 
i 3 1 Year*, at 5, and 
16 pp. Cent. Simple 
\ Interest. 

) Years. 5 



TAB LE II. 

Shewing the 
Rebate of 1 Dol- 
lar, op 1 Pound for 
31 Year 8, at 5 an<< 
6 pr. Cent. Sim- 
ple Inter eat. 



% 



1 


105 


2 


110 


3 


MS 


4 


1-20 


5 


1-25 




_. 


6 


1-30 


7 


I'35 


8 


1-40 


9 


1-45 


10 


1-50 


, 


— — . 


11 


1-55 


12 


1-60 


IS 


r-65 


14 


1-70 


15 


1-75 


» ■ 


■■"■■— 


16 


1-80 


17 


1-85 


18 


1-90 


19 


p95 


20 


fc-00 


■ '- 


_ 


21 


2-05 


22 


2*10 


23 


2-15 


24 


5-20 • 


25 


2.25 


26 


2-30- 


27 


2-35 


28 


2-40 


29 


' 2-45 | 


30 


2-50 


31 


2.55 1 



2-26 
2*321 



9523SO 
90909 1 
•869565 
•833333 
'800000 



•769230 
•740740 
•714286 
689655 
•666666 

•645161 
•625000 
•606060 
•588235 
•571448 



•555555 
.540540 
•526315 
•512820 
•500000 



487804 
■476190 
'465116 
454545 
444444 



434781 
'425532 
•416666 
■408163 
400000 
•393157 



892857 
847457 
806451 
769250 

735294 
704225 
675675' 
649350 
625000 

602409 
581395 
561797 
543478 
526315 

510204 
495049 
•480769 
467289 
454545 

442477 
431034 
•420168 
409836 
400000 

390625 
381679 
73134 
364963 
357143 
34965C| 



TABLl, i ll. 

Shewing the Amo.i 
of 1 Dollar, pr 1 Pour 
for Months at 5, and t-l 
pr. Cent. Simple Inte\ • 
cat. 



Mont 



1-00416 
100833 
I 01249 
1-01666 
1*02083 

1 02499 
1 02916 
103S33 
1-03749 
1-04166 
1-04583 
1 05000 



1-005 
1-0 l€i 

1.015 
1-020 
1-025 

1-030 
1 055 
1-040 
1045 
1-050 
1*055 
1-060 



1 
2 
3 

4 
5 

6 

7 

8 

9 

10 

11 

12 



Shewing the Rebau ' 
or Present worth of l [ 
Dollar, or 1 Pound, for 1 
Months dicouniing at? 
5, and 6 per Cent. SiVw-T 
file Inter eat. Y 



Monti.* 



99585 
-99173 
■98766 
98361 
97959 

•97561 
•97263 
■96772 
•96387 
•96006 
95617 
95238 



•99502 
.99009 
•98522 
•98039 
•97560 

•97087 
•96628 
•96153 
•95690 
•95238 
94786 
•94339 



1 
2 
3 
4 
5 

6 
7 
8 
9 
10 
11 



$ 



£ 31 2 55 I 2-86 -393157 -34965C| 95238 -943^^ l * ^ 



15* LOGARITHMIC* • 

CoMSTUCTION OF T ABLE 8. &C. 

The two first give the Amount, and Rebate, ot Presekt 
Worth of 8 1, or £\ y from 1 to 3! years inclusively, at 5 and 
6 pr. cent. Simple interest. They are calculated by making 1 
dollar, or 1 pound the Principal in the first, and 1 dollar or 
I pound the Amount in the second ; or dividing unity by the 
srveral numbers in the first Table, gives the numbers in the 
second Table. The third and fourth Tables are of the same 
nature with the first two, and are therefore subject to the same 
principle of construction. 

Application and Use of the precbe&ing Tables. 
CASE I. 
To find the amount of any given turn/or year* and monthly 
at 5) and b fir. cent, Simple Interest. 

RULE. 
To the Tabular number found in Table I, under the given 
rate and opposite the time in years, add the number found on 
the right of the decimal floint in, Table HI, under the given rate 
and opposite the months, and multiply this sum by the princi- 
pal ; and the product will be the amount sought. 

EXAMPLES. 

1. What will 8 100 amount to in 7 years and 8 'months, at .6 
pr. cent pr. annum, Simple Interest ? 
Tabular number = 1.42 = amount of gl for 7 years. 
Do* .04a = do. for 8 months. 



1.460 
Multiply by .100 = principal, 

8146.000= Ans. 



2. Required the amount of 8318 50 cents* for 5 years, at 4 
per cent per annum. Ans. 414 05cts. 

3. What will 8753 215 cents amount to in 4 years and 7 months,! 
at 5 per cent per annum ? Ans. 8925 86c* 3^m. 

4. Required the amount of 1121. 10s. for 3 years and * 
months, at 6 per cent. Ans. 1351. lis. 4d. 

5. Required the amount of 1801. 8s. for 1 1 months, at 6 pei 
cent per annum* Ans. 1901. 6s. 5d. 1 qr. 

CASE II. 
To find the Interest of any given sum for years and monthg^ 
at 5 and 6 fier cent* 

RULE. 
1 . Find the amount as before, from which subtract the priiuji* 
palj and the remainder will be the interest. 



ARITHMETICS. 121. 

OR 

2. Multiply the number found on the right of the decimal " 
point in the table, by the principal, and the product is the inter-* 
est required. 

EXAMPLES. 

1. What is the interest of $400 forbora 3^ years, at t pe> 
cent per annum ? 

Tabular number for 3 years = 1.20 
Do. for 6 months. .02499 



Multiply by 


1.22499 
400 




Subtract 


' 489.99600. 
400. 


Amount. 
Principal. 


Answer* 


889.99600 = 


= 890. fere.- 



2. What will be the interest of 8210 35 cents, for 9£ years, at ' 
6 per cent per annum? Ans. 8123 05c. 4.75 m.* 

3. What will be the interest on £45 10s. for 1 year and T 
months, at 6 per cent ? V Ans. £4 6s. 5d. 1 qr. 

4. Required the interest of £ 896 1 5s. for 7 months, at 5 per' 
cent per annum. Ans. £27 2s. 1 id. 3 qrs. 

CASE III. 
Tojind the Rebate or Present worth of any given sum for 
years and months. 

RULE. 
Multiply the Tabular number under the given rate and oppo- 
site the time by the principal, and the product will be the fire- 
sent worth. 

EXAMPLES. 

1. What is the rebate, or present worth of 8 100 due 1 year 
hence, discounting at 6 per cent per annum ? 
Tabular number = .943396 
100 



894.33.9/ T Ans* 

2. Wfcat is the present worth of g 1 80 50 cents, due 5 years 
hence, at 5 per cent per annum ? Ans. 8138 84c. 6m. 

3 How much ready money will pay a debt of £ 1 12 10s. due 
3 years hence, discounting at 6 per cent ? £95 6s. lid. 

4. How much ready money is equal in value to £315 8s. duo 
7 months hence, allowing 5 per cent discount ? 

Ans,jr306 15s. 4d. 

Note. When the discount is required, subtract the present 
worth from the principal, and the remainder is the discount'. 



i 



lea TSOCARfTHMtCfc 

5. What is the discount on£*$60, due 7 years hence, at 6 per 
cent per annum? £Ans. 147 17s. ftd. 



COxMPOUND INTEREST BY DECIMAL?.; 

RULE. 
i. Find the amount of gl, ov£\ y for one year at the given rate 
per cent. 

2. Involve the amount, thus found, to such a power, as is de- 
noted by the number of years, and multiply this power by the 
principal, or given sunt* and the product will be the amount re- 
quired. 

3. Subtract the principal from the amount, and the remainder 
-will be the.itfcrfrf* 

EXAMPLES. 

I. What is jthe compound interest of £500 for 4 years, at 5 
jper cent per annum ? 
^The amount of 11. /or J year 7= 1.05 and 
llW'X 5Q0>=? ^07.753125 = the amount. 

/500 

107.75312.5 *=,£107 15s. Ojd. interest required. 
.2. What U the amount of #7>60 LOs. for 4 years, at 4 per ct. ? 

Ans. 8891. 13s. 6id. 

«f Table qf the amount of&\..or \ Lot 6 fier .cent per annum, for 

months. 




When the given time consists, of years and months, seek the 
amount $f %\ &c. in the table for years, and the amount of g 1 
Icc.in the foregoing table for the, months, and the continual pro- 
duct of these tabular number* into vtfce principal,. will. give the 
amount requiret 1 . 

Note. Subtract the principal .from the amount, and the re» 
mainder is. the compound interest? 



ARITHMETICS, 



1^? 



1. Required the ahioirot of ^480 for 5 years land 6 mo0tKs,at 
6 per cent per annum compound interest. 

Tabular number of \L for 5 years = 1.334285 
Do. " for 6 months = 1.02$56O 



1.77*82931 

480 = prinl,. 



An*. /"66 1. 23*1 fee. 

2. What will 8100 amount to, forborn 7 years and 10 months, 
it 6 per cent pdr annum ? Ans. 8157 82c. 3m. ' 

3. What is the compound interest of 82 10 bQ cents, for 3 
Jeers* at 6 per dent I Ans. 829 48c, 7m. + 

4. What is the compound interest of 8QJ. 4s. for 9 yeara and 
4 months, at 6 per cent per annum ? Ana. 471. 10s* Cf d. 

Another method of computing Compound Interest for years^ 
months, and days. , - 

RULE. 

To the Logarithm of the principal, add the several logarithms 
answering to the number of years, months, and days, found in 
the following tables, and their sum will be the logarithm of the 
amount required. 

Lagarithmick Tabhesy at ^ jttr cent per annum, for years, months 

and days* 

iSvtatafc i din i Jit i aTu taTu I Ait 1 flat i?l~ iJiKi itf in lain 1 3*i i jtn> i jffin i il ■■ & 
i F. J dec. /**». J F. I dec. fits. | Y. J dec. fits. \ V. | d&Jits.j: 

U 

" 3 
If 

( 6 

ii r 

II 8 

i 1 o 

(110 
'»¥(fiitrfta* i^twjfafc jdlafcHflnt f fTah *fr ,%, «V,rfafAh.taijiwfl>fc i^iitfiii f4n "*■•- «£ 



,025306 ; 


11 


,278366 


21" 


,531426 • 


31 


,73458*6 & 


,050612 


12 


,303672 


22; 


,556732 


32 j 


,809792 & 


,075918 


13 


,328978 i 


23 


,582038 


33 


,835098 S 


,101224 


14 


,354284 


24 


,607344 


34 


,860404 a 


,126530 


15 


,379690 


25 


,632650 , 


35 


,885710 6 


,151836 " 


16 


,404896 


26 


,657956 


36 


^911016 & 


,177143 


17 


,430202 


27 


,683262 


37 


,936322 f 


,202448 


18 


,455058 


28* 


,708568 


"38 


,96*628 ^ 


,227754 


19 


,480814 


29' 


,733974 


39 


,986934 ^r 


,253060 


20 


,506120 


30 


,759380 


40 


1,01224 f 



IJ24 



i-OGARITHMICK 



[Af. | dec fits. M. J dtc./ics. \ Af. \ drc. fata. | M, dec. fits.] 



i 



,002160 



2 \ ,004321 
,006466 



I I V ?*tt*aft4~ftaft«-ttft*6«ft+*a«a#tte0« *"«&tt«ft«ttttfr#Oft**ft*«? 9 



/I. | 



4 ,006600 
* I ,010724 
6 | ,012337 



«-*** «»^»«^#4««M<r »o*»e>» »»***€>»+# »**»»*+*»« 



*■! 



•ii 



1 


,0000? 1 


9 


2 


,000143 


10 


3 


,000215 


11 


4 


,000287 


12 


5 


,000358 


IS 


6 


,000429 


14 


7 


,000500 


15 


8 


,000571 


16 



,000642 
,000713 
,000785 
,000857 
,00C928 
,000999 
,000107 
,001142 



7 | ,014940 | 10 

8 J, 017033 J 11 

9 J ,010116 | 



,0211891 
,023252 ( 



^•1 



\V. | 



17 
18 
19 
20 
21 
22 
23 
24 



,001212 
,001284 
,001355 
,001426 
,001497 
,001568 
,001639 
,001710 



25 
26 
27 
28 
29 
30 
31 



,00178 1( 
,001852(| 
,001 925 < 
,001994^ 
,002065 < 
,002136^ 
,002207 < 



iir iHli, fji« *^»- «j>». .A*. »^«. .<.fet «i» .A». .A». «J«- »J«. .^.. i^» t 
TffiT^^Jf' t^tH^JJ* l«|V p 1^* '9* ™ff " V *^ *™' ^^** ^^f t^*» t^W 

1. Required the amount of 8436 50 cents, for 3 years, 8 
months and 12 days. 

Log. of principal = 2. 6399842 

Add Log. of years = 0. 0759 1 80 

Do. of months = 0. 0170330 

Do. of days = o. 00085 70 



Ans. &541 75 2. 7337922 Amount requirM 
2. What will 8175 amount to, in lo years and 5 months, at 6 
per cent per annum I Ans. 8321 28 cts. 

CASE II. 

The amount, rate, and time given, to find the firincifiqL 

RULE. 

T. Divide the amount of the given sum by the amount of £1, 

or* £1, for the given time, and the quotient will be the principal ; 

OR 

*2. Multiply the present worth of £1, or £\, for the given 
number of years, at the given rate, by the amount ; the product 
will be the principal. 

EXAMPLES. 

1. What principal at 6 per cent per annum, will amount to 
£757.4856, in 4 years ? 

By Table I. 

By Table II. 
Multiply by the present worth > 757*4856 
of 8 I for 4 years, at 6 per cent. 5 '792093 

Ans. 8 600 principal. 

2. What principal at 6 per cent, per annum, will amount tp 
£757 9s. 8-Jd. in 4 years ? Ans. 600/. 



ASITHMETIC&. 13&< 



ARITHMETICAL PROGRESSION. 

Any rank of numbers) increasing by a common excess, and 
decreasing by a common difference, is said to be in Arithme- 
tical Progression. 

C 14.12.10.8.6 &c? r. 

I 7. 6.5.4.3 kc. i DeteadmgieriM. 

The numbers which form the series, are called the Terms of 
the progression ; the first and last terms of which are called the 
Extremes. 

Any three of the five following terms being given, the other 
two may be readily found. 

1. The first term. 

2. The last term. 

3. The number of terms. 

4. The common difference. 
& The sum of all the terms. 

PROBLEM I. 
The first term, the last term, and the number of terms being 
given, to find the sum of al) the terms. 

RULE. 
Multiply the sum of the extremes by the number of terms/ 
and half the product will be the answer. 

EXAMPLES. 

1. The first term of an arithmetical progression is 5, the last 
term, 60, and the number of terms 12, required the sum of the 
series. 5 -J- 60 X 12 -u 2 = 380 Ans. 

2. The first term of an arithmetical progression is 3, the last 
term 1 12, and the number of terms 18 ; required the sum of tho 
series. Ans. 1035 

3. How many strokes do the clocks of Venice, (which go to 
24 o'clock,) strike in the compass of a day ? Ans. 3Q0 

4. Suppose a man lay up 1 cent the first day of the year, 2 
cents the second, and 3 the third day, and so on in arithmetical 
progression, every day increasing 1 cent ; how much will he 
have saved at the year's end ? Ans. 2667,95 cents. 

5. A merchant bought 100 yards of cloth in arithmetical pro- 
gression ; he gave 5 cents for the first yard, and 1 dollar for. 
the last, what did the cloth amount to ? Ans. g52,50 cts. 

6. If 100 stones be placed in a right line, exactly a yard asun- 
der, and the first a yard from a basket, what length of ground 
will that man go, who gathers them up singly, returning with 
them one by one to the basket ? 

Ans. 5 miles, 233 rods,. 2 yards* 

17 



l& > «>GAR*THMICk 

PROBLEM II. 
The first term, the last term* and the number of tetms giy£n> 
to find the Common Difference. '■ 

RULE. 
Divide the difference of the extremes by the number of term* 
less by i, and the quotient will be the Common difference re- 
quired. 

EXAMPLES. 

1. If the extremes be 3 and 19, and the number of terms 9, 
what is the common difference, and the sum of the whole series.? 

19 

Extremes J ! J J[ 

— 22 
9 — l = 8)16(2 common difference 9 

16 - — 

— i) 198(99 sum of series. 

2. A man had 10 sons, whose several ages differed alike, thA 
youngest was 4 years old, and the oldest 40 ; what was the 
common difference of their ages ? Ans. 4 years. 

3. A man travels from Manchester to London in 6 days; 
every days journey was greater than the preceeding one, by a 
common excess ; he traveled 20 miles the first day, and 40 miles 
the last ; what was the common increase of each succeeding 
day's, journey and the distance from Manchester to London ? 

Ans. daily increase 4 miles, and the distance of journey 
180 miles. 

PROBLEM III. 
The two extremes and the common difference given, to find 
the number of terms. 

RULE. 
Divide the difference of the extremes by the common differ- 
ence, and the quotient, increased by 1, is the number of terms 
Required. 

EXAMPLES. 

1. If the extremes be 3 and 19, and the common differejicej2> 
what is the number of terms ? 

19 — 3-5-2 = 8+1 = 9 Ans. 

2. A man, going a journey, travelled the first day 3 miles 
and the last day 43, and increased his journey every day 5 miles ; 
how many days did he travel ? Ans. 9 days. 



ARITHMETIC^. • ,1,3? 



GEOMETRICAL PROGRESSION. 

Any series of numbers are in Geometrical Progression* 
.when the several terms increase bya coinmon multiplier, or de- 
crease by a common divisor ; — Thus, 3, 6, 12, 24, 48 &c~ is a 
series in geometrioal progression, increasing by the common. 
Multiplier 2 ; and 81, 27, 9, 3, 1, 8cc. is a series in geometrical, 
progression, decreasing by the common divisor 3. 

The number, by which the series is constantly increased, or 
diminished, is called the Ratio. 

PROBLEM I. 

Given the first term, the last term, (or extremes) and the ra** 
tio, to find the sum of the series. 

RULE. 

Multiply the last term by the ratio, and from the product sub- 
tract the first term, and the rem ainder, divided by the ratio less 
1, will give the sum of all the terms of the series. 
examples. 

1. The first term of a series in geometrical progression is o«. 
the last term 531441, and the ratio 3 ; required the sum of all 
the terms. 

The series is 3,9,27,8 1,243,729,2 187,6561, 19683,5904$, 
177147,531441. Then, 

531441 X 3)— 3(-~3 — 1 c= 797160 Ans. 

2. The extremes of a geometrical progression are 1 and ' 
65536, and the ratio 4 j what is the sum of the series ? 

Ans. 87381 
PROBLEM II. 
Given th,e first term and the ratio, to find any other term as- 
signed, 

CASE I. 
When thcjirst term of the series and the ratio are equal,* 

1. Write down a few of the leading terms of the series, and 
place their indices over them, beginning with an unit, or 1. 

2. Add together such of the most convenient indices, as wilL 
make up the entire index to the sum required. 

* Note. When the first term of the series is equal to the ra« 
tio, the indices must begin with an unit, and the indices added - 
roust make the entire index of the term required ; but if 
the first term be greater, or less than the ratio, the indices must 
begin with a cypher, and the indices added must make an index 
less by I than the number, expressing the plaqp of the t^rni 



it* LOGARITHMIC* 

3. Multiply the terms of the geometrical series belonging t^ 
those indices, continually together, and the product will be the 
term sought. 

EXAMPLES 

•If the first term be 2 and the ratio 2, what is the Istb 
ftrm I 

ThtM 5 l>2 >*> 4 » 5l 6 Indices. 
in ' {2,4,8,16,32,64 Leading terms. 
Then 4+S T 6a= 15 the index of 15th. term, and product 
df 16, X 32 X 64, =• 32768 term required. 
% A merchant bought 22 hhd.of wine for 2 mills for the first 
&hd. 4 for the second, 8 for the third, and soon in duplicate pro- 
portion geometrically ; what did the whole amount to at that 
rate I 

The 2 2d. or last term is 4194304. 
v Then 2X4194304—2 

■ mi =. 8398606, the sum of all the terms 

2—1 
(by PaoB. I.) = $ 8398,60t. 6m. Ans. 

3. A labourer agreed to work one whole year for a rich mi- 
dfer, to receive no other reward than 3 farthings for the 1st* 
month, 2£d. for the 2d. month, 6f d. for the 3d. month, and so 
on, in triple proportion geometrically ; what did his wages 
amount to in one year, and what was the average price of each 
day's labour I Ans. His wages for one year amounted to 
£"850 7s. 6d. and the average price of each day was £ 2 7s. 6d. 
CASE II. 
When the first term of the series and the ratio aTe different* 
that is, when the first term is either greater or less than thfe 
ratio. 

RULE. 
1. Write a few of the leading terms as before, and begin 
their indices with a cypher. 

2.' Add together the most convenient indices to make an in 
dex, less by 1 than the number expressing the term sought. 

3. Multiply the terms of the geometrical series together 
belonging to those indices, and make the product a dividend. 

4. Raise the first term to a power whose index is one leas 
than the number of terms multiplied, and make the result a di- 
visor. 

5. Divide the dividend by the divisor, and the quotient will 
be the term sought. 

•» » - ■ ■ 

* Note. See this principle explained by logarithms,— page 44 



ARITHMETICS. *» 

EXAMPLES. , 

1. If the first term of a geometrical series be S, and the nm 

3, what is the 1 1th term ? 

Thn* S°> *> 3 > 3 » 4 > Indices. 

Uf {5,15,45,135,405, Leading terms. 
And 1+2 + 3+4 ae 10 the index of the 11th. term. 
15X45X135X405 = 283943125 

=2263545 the 1 ltlxc 

53,-. 125 [term required. 

Here the number of terms multipled are 4 ; therefor© the* 

Jst. term raised to the 4th. power less by 1 is the 3d. power, or 

cube of 5 = 125 the divisor. 

2. What debt can be discharged in a year, by paying 2 cents 
for the first month, 8 cents for the second, 32 cents for the 
third month, and so on in quadruple proportion, for each 
month ? Ans. 8 111848 10 cts. 

3. An ignorant horse jockey being employed to purchase a 
number of horses for shipping, very readily agreed with a gen- 
tleman, well skilled in numbers, for 28, upon condition that he 
should give 1 cent for the first horse, 5 for the second, 25 foT 
the third horse, and so on in quintuple proportion to the last 
horse ; what did they come to at that rate, and how much did 
they cost per head ? 

Ans. the horses came to g 307708728652954101,56 cts. and 
the average price was, $ 10989597451855503,62 cts. 7Jm. per 
licad. 

4. What will a horse cost, computing his worth in geometri- 
cal progression by the nails in his shoes, at a farthing for the 
first nail, 3 farthings for the second, and so on in triple propor- 
tion to the last, or 32d. nail ? Ans. £9651 14681693 13s. 4d. 

5. A young man skilled in numbers, agreed with a farmer to 
work for him 1 1 years, without any other reward than the pro- 
duce of one wheat corn for the first year, and that produce to 
be sowed the second year, and so on from year to year, till the 
end of the time, allowing the increase to be in a tenfold propor- 
tion ; what quantity of wheat is due for such service, and to 
what does it amount, at 8,150 per bushel ? 

Ans. w 226056! bushels, allowing 768 whqat corns to make a 
pint; and the amount is £339084 18c. 24m. 

6. What will be the value of a 64 gun-ship, reckoning 1 pen- 
ny for the fir6t gun, 2 pence for the second, 4 pence for the 
third gun, and so on to the Jast, in duplicate proportion ? 

Ans. £786 1435640456465 IS. i d. 



130 LOGARITHMICK 

7. Suppose America should agree to build 144 ships of the 
Jthe for Great Britain, at the rate of but 1 farthing for the first 
ship, 2 for the second, 4 for the third, and so to increase, in a 
duplicate proportion to the last ; what would they all amount to 
at that rate, and how many globes of standard gold, equal in 
magnitude to the earth we inhabit, could be formed from the 
mass, allowing a cubic inch of gold to be worth £53 2s 8d. ? 

1 the 9th Ship=: 256 the 18th Ship = 131072 

131072 

262144 
9175,04 
131072 
393216 
65536 131073 
131072 ,' ■ ■/ ■ 




17179869184 

The 36 Ship 34359738368 

34359738368 



274877906944 
206158430208 
103079215104 
2748779Q6944 
103079215104 
240518168$T6 
309237645312. 
* 171798691840 

103079215104 
137438953472 
103079215104 
■ ■ ' ' ■ > ■ 

1180591620717*11303424 
The72d. Ships 2361183241434822606848 



As multiplying the index of any term in a geometrick series*, 
by a given number, gives the index for that fiower of the term, 
denoted by the multiplying number ; therefore, as 72 X 2 = 144, 
raise the 72d term to the second power, that is, multiply it by 
itself, and that product by the ratio, the last product will be the 
amount of the last term, which by (PROB. 1 .) will give the sum 
of all the series : as follows. 



4BIJHMETICX *M 

Tf)fe72d. Ship a 2361183241434822606343 
2361183241434822606848 



18889465931478580854784 
9444732965739290427392 
18889465931478580854784 
14167099448608935641088 
141670994486089356410880 
4722366482869645213696 
47223664828696452 1 5696 
1 888946593 1 478580854784 
94447329657,59290427392 
7083549724304467820544 

444732965739290427392 * 

2361 183241434822606848 
9444732965739290427392 
4722366482869645213696 
7083549724304467820544 
188.89465931478580854784 
. 236I18324143482260G848 
2561183241434822606848 
14f67p99448608935641088 
. 7083549724304467820544 
-4722366482869645213696 

— m I" 

5575186299632655785383929568162090376495104 Ship. 
1 1 15037259926531 1570T67859 136324180752990208 ss 144 

4)22300745198530623141535718272648361505980415 T. Sum. 
12)5575186299632655785383929568162090376495103 
30)464597191636054648781994130680174198041258— 7d. 

£232298595818O27324390997O6.53400870?9O2O12 _ 18s. 7d. 

Which is the exact amount of the whole number of ships ; and 
ia»computed according to the conditions of the question. 

Now to reduce this to solid gold, divide the amount by 53J. : 
2s. 8d. (or 12752d.) and it will give 

,4372004174743212871580376161734358302 the number of solid 
inches the mass would contain. And in order to compare {his 
«mass with the solidity of the earth, this also must be reduced to 
inches. The number of solid miles contained in thfc*eartlr(see 
page 99) is 263858 1 49 1 20, which multiplied by 254358&8873feoOO 
the number of cubic inches in 1 mile, gives 
66646680917786616312320000 solid inches $ by whicfe 'divide 
the number of sofid inches in the whole mass, and the quotient 
will be the number of globes, (equal in magnitude to the earth,) 
' contained In the mass, which is the Answer required s as follows. 



_132. LOGARITHMICK 

SOU. "««*J SOXIB INCHES IN THE WJIOtE MASS. 5**™"* « 
ip THB KARTH. ) £ GLOBES. 

666466809 177 8-) Ans. 

661631232QOOQ)4372004l7474321287158037616l734S583Q2(655997285ir 

S9988008550671969787392000Q 



373203319676015892841176161 
333233404588933081561600000 

399699150870828112795761617 
333233404588933081561600000 

6646574628 1 89503 1 234 1 6 1 6 1 73 
599820128260079546810880000 

648373345588707755307361734 
59982012826007954681088000O 

485532 1 732862820849648 1 7343 
466526766424506314I8624000O 

19OO54O68516757707795773435 
133293361835573232624640000 

567607067811844751711334358 
53317344734229293049856000Q 

344736204695518212127743583 

3332334045889330ai 56 1600000 

t 

115028001065851305661435830 
66646680917786616312320000 

483813201480646893491158302 
• 466526766424506314186240000 

_^_ * 

And something over. 

Tims we have proved, what at first might not seem to be easy of be- 
lief, that the whole amount in Sterling Money is twenty three thousand 
two hundred and twenty nine miliions of millions of millions of millions of 
millions of millions, eight hundred fifty nine thousand five hundred and 
eighty one millions of millions of millions of millions of miliions, eight- 
hundred two thousand seven hundred and thirty two millions of millions 
of millions of millions, four hundred thirty nine thousand and ninety nine 
millions of millions of miliions, seven hundred six thousand five hundred 
and thirty four millions of millions, eighty seven thousand and ninct£.nine 
millions, nine hundred two thousand and twelve pounds, eighteen i&H- 
Tings and seven pence* * V 



ARITHMETIC^. 

And the number of solid globes of gold, equal to tke earth we 
inhabit, is sixty live thousand five hundred and ninety nine mil- 
lions, seven hundred twenty eight thousand five hundred and 
seventeqp. 



POSITION. 

Position is a rule, which by false, or supposed numbers tak- 
en at pleasure, discovers the true one required. 
It is divided into two parts, Single and Double. 

SINGLE POSITION. 
Single Position teaches to resolve such questions, whose re- 
sults are proportional to their supposition ; and is when the 
proportions of the required number are implied in the conditions 
of the question, 

RULE. 

1. Take any number and perform the same operations with 
it, as are described to be performed in the question. 

2. Then say ; as the result of the operation ; is to the given 
sum in the question : : so is the supposed number : to the true 
one required. 

examples; 

1. A Schoolmaster being asked how many sc hollars he had, 
said, if I had as many more as I now have, half as many, and one 
fourth as many, I should then have 99 ; how many scholiars 
had he ? As 1 10 : 99 : : 40 ; 36 Ans. 

Suppose he had 40 36 

as many =40 18 

\ as many = 20 9 

£ as many =10 — 

— Proof 99 

Result 110 

Or,— As 110: 40: : 99 : 36 the Ans. 

2. A person after spending one third and one fourth of hi^s 
money* had £ 60 left ; what had he at first ?* N Ans. g 144. 

3. What number is that, a sixth part of which exceeds an 
eighth part of it by 20 ? Ans. 480 

4. What sum of money is that, whose third part, fourth part, 
and fifth part, added together, amount to 94 dollars ? 

Ans. g 120 ' 

5. In a mixture of corn and oats, £ of the whole plus 25 bush- 
"* els was corn, -J. part minus 5 bushels was oats ; how many bush- 
els were there of each ? Ans. 85 of corn, and 35 of oats. 

18 



\ 



r 7 



LOGARITHMICK 

6. What number is that, from which if 5 be subtracted, f of 
the remainder will be 40 ? Ans. 65. 

7. Two travellers, A. and B. 360 miles apart, travel toward* 
each other tilflhey meet. A's progress is 10 miles in an hour, 
and B's 8 ; how far does each travel before they meet ? 

Ans. A. goes 200 miles, and 13. 160. 

8. If a certain number be divided by 12, the quotient, dividend 
and divisor added together will amount to 64 ; wBat is the num- 
ber ? Ans 64. 

9. A. man spent one tfcircl of hi$ Bfe fa England, one fourth 
of it in Scotland, and the remainder of it, which was 20 years* 
in the United States ; to what age did he live ? 

Ans. to the age of 4$: 

DOUBLE POSITION. 

Double Position teaches to resolve questions' by making 
two suppositions of false numbers. 

RULE. 

1. Take any two convenient numbers, and proceed with each 
according to the conditions of the question. 

2. Find how much the results are different from the result in 
the question. 

3. Multiply the first position by the last errour, and the la^t 
position by the first errour. 

4. If the errours are alike, divide the difference of the pro- 
ducts by the difference of the errours, and the quotient will be 
the answer. 

5. If the errours are unlike, divide the sum of the products 
by the sum of the errours, and the quotient will be the answer. 

Note. The errours are said to be alike, when they are both too 
great, or both to© small ; and unlike, when one is too great ami 
the other too small. 



ARITHMETICS IS* 

EXAMPLES. 

1. The ages of 4 persons amount together, to 109 years, A i* 
7 years older than B, and C is 10 years younger than A, and D 
;h | as old as A ; required the age of each. 

1st. Suppose A's age =a 40 ad. Suppose A. s= 30 

B's " = 33 B. = 23 

C's « =t 50 C> = 20 

D's « = 24 D. = 18 

127 — 9i 

—109 109 

1st. errour 18 2d. errour 18 

The errours being unlike or one too great, and the other too 
small. 

Pos. Err. 
40 30 



Therefore 



X 



** 



AV » 35 
B J s s= 28 

C's = 25 

18 18 (JO'S = 21 

30 40 

— Proof 109 

540 720 



18+ 18 =* 36)1260(35 = A*s agft. 

2. Three merchants enter into partnership with a stock of 
$ 1 140, A put in a certain sum, B put in one third as much as 
A and 8 50 more, and C put in twice as much as B, together 
with a fifth of what A put in ; what was each one's respective 
•hare in the stock ! 

Ans, A put in g 450, B g 200, and C S 490 

3. The ages of two persons A and B are such, that 7 veapi 
ago, A was three times as old as B ; and 7 years hence, A will 
be twice as old as B ; what are their respective ages I 

An«. A 'sage is 49, knd B's21 years 

4. Three persons A, B, and C, purchase ahorse for 100 dol- 
lars, but neither is able to pay for the whole : the payment 
would require 

The whole of A's money, together with half of B's ; or 
The, whole of B's, with one third of C's ; or 
The whale of C's, with one fourth of A's ; 
How much money bad each ? 

Ans. A, had 8 64., B g 72, and C g 84 

5. Tire sum of the distances which 3 persons travelled, is 62 
nujes ; A travelled 4 tildes as^ far as. C, added to twice thedia* 



136 LOGARITHMICK 

fence that fi travelled, and had C travelled IT times as far as he 
did, he would then have travelled 3 times as far as B, added to 
twice the distance that A travelled ; required their respective 
distances ? Ans. A travelled 46 miles, B 9, and C 7 miles. 



PERMUTATION OF QUANTITIES. 

The permutation or variation of quantities is the 
showing how many different ways the order or position of any 
given number of things may be change^. 

To find the number of permutations or changes, that can be 
made of any given number of things, all different from one 
another. 

RULE. 

Multiply all the terms of the natural series of numbers, from 
one up to the given number, continually together, and the last 
product will be the answer required. 

EXAMPLES. 



1. How many changes can be made of the fl 

fetters in the word and ? 



Proof 



\ 



and 
a d n 
n a d 
n d a 
dan 
d n a 



I X 2 X 3 =- 6 Ans. j 5 

i« 
2. How many changes can be rung on 12 bells ? 

Ans. 479001600 
S. How long could a family of 9 persons vary their position. 
c£ dinner ? Ans. 994 years 80 days. 

4. How many changes can be made (in position) of the 8 
notes in musick ? Ans. 40320 

5. How many variations may be made of the letters in the 
English alphabet ? Ans. 403291461 *2 66056355 8 4000000 



CONSTRUCTION OF THE FOLLOWING TABLES 
BELONGING TO COMPOUND INTEREST- 

The construction of these Tables by logarithms, will 
be best understood by the following proposition. Viz. 

Between two numbers given, to find any number t>f mean 
proportionals required. 



ARITHMETICS I3t 

RULE. 

1. From the logarithm of the greater number subtract the 
logarithm of the less, and divide the remainder by the number, 
of means increased by 1 

2. Add the quotient to the logarithm of the less, number, and 
the sum will be the logarithm of the 1st mean proportional re- 
quired. 

3. To the logarithm last found, add the said quotient, and the 
sum will be the logarithm of the second mean proportional ; 
and thus proceed, always adding the said quotient to the loga-, 
rithm of the last proportional found, as far as the question re* 
quires. 

examples. 
Required to find, between 16 and 64, 5 xpean proportionals. 
Logarithm of 64 = 1-8061800 

Do. #f 16 = l-204i 200 



The difference ±= 06020600 

To \ part for 5 means = 0-1003433 

Add logarithm of 1 6 1 -204 1 200 , 

1 st. Mean proportional = 20- 1 58 = 1*3044633 

To which add said*quotient = ' 0-1003433 

2d. Mean proportional =* 25-398 «= 1 -4048066 

Add quotient 0-1003433 

3d. Mean proportional =32 = 1-5051499 

Add quotient 0*1003433 

4th. Mean proportional = 40-3 1 7 = 1 .6054932 

Add quotient 0-1003433 

5th Mean proportional = 50*796 =• 1.7058365 

For the construction of the first two following tables are sev- 
eral methods used. We shall mention only that which is irtost 
easy and expeditious ; which is by logarithms. 

For the first Table thus : Find the amount (as already 
taught) of 1 dollar or 1 pound for 40 years at the given rate per 
cent, and between the logarithm of the amount and the loga- 
rithm of the rate find 40 geometrical mean proportionals, by the 
last proposition ; and these will be the logarithms of the num- 
bers in the first Table? Or, 



i 



**• LOGARITHMICK 

If we add the logarithm of the rate continually to itself, it wilt 
give the same result ; Thus, adding the logarithm of the rate 
to itself, gives the logarithm belonging to the second year, 
*nd to this sum adding again the logarithm of the rate, gives 
the rogarithm of the number belonging to the third year, &c. 
Or, if ^>u multiply the logarithm of the rate, by the numbers 
1, 2,3, 4, 5, 6, &c. gives the logarithm of the numbers belong- 
ing to those respective years ; 

' And for the numbers in the second Table find the Arithme- 
tical Complements of the logarithms of the numbers in the first 
Table, and you will have the logarithms of the numbers in the 
second Table. 

The Logarithmical differences of 045, or 05, or 06,(being the 
rates here used, minus unity,) and the numbers in Table I, are 
the logarithms of the numbers in Table III. 

If, from the logarithms of the numbers in Table III. you 
subtract the logarithms of the numbers in Table I. you will 
bave the logarithms of the numbers in Table IV. 

And their Arithmetical Complements are the logarithms of 
th* numbers in fable V. 



ARITHMETIC^ 



l J* 



IfABLK I. Shewing the amount 
of% l,or£\frpm 1 year to 40. 



TABtE II. Shewing the 
present value of% 1, or £i 
due at the end of any num- 
ber of years, from \ to 40 



Y. 


4f fir. ct.\5 fir. ct. 


6 fir. cs. 


4\ fir. ct 


Sfir.m. 


f>fir. ct. 




\ 


L 045000 


1-050000 


'1*060000 


•956931 


J5I4381 


943396 




2 


t 092025 


1- 102500 


1- 123600 


•915730 


907030 


•88999J& 




S 


M41166 


1-157625 


1-191016 


•876297 


863838 


•839619 




4 


1-192518 


1*215506 


1-262476 


•838561 


822702 


•792093 




5 

■ 

6 


1*246181 


1-276281 


1-338225 


•802451 


783526 


•747258 




1.302260 


1-340095 


1*418519 


•767896 


•746215 


•704960 




7 


1*360861 


1-407100 


1*503630 


•73482fc 


710681 


•665057 




6 


1-422100 


1-477455 


1*593848 


•703185 


676839 


•627412 




9 


1*486095 


1-551328 


1-689478 


•672904 


6446C9 


•591898 




10 
11 


1*552969 


1-628894 


1-790847 


•643928 


613913 


•558394 


k 


1-622853 


1-710339 


1-898298 


•616199 


^584679 


•562787 




12 


1-695881 


1-795856 


2012196 


•589664 


■556837 


•496969 




13 


1-77219* 


1*885649 


2132928 


•564271 


53oS21 


•468839 


N 


14 


1-851944 


I -979931 


2*260903 


•539973 


•505068 


•442300 




15 


1-935282 


2078928 


2*396558 


•516720 
•494469 


•481017 


•417265 




16 


2*022370 


2* 182874 


2-547271 


•458311 


•393647 




17 


2- 11 3376 2-2920 18 


2-692772 


•473176 


•436297 


•371364 




18 


2*208478 2*406619 


2-854339 


•452800 


415521 


350343 




19 


2*307860|2-526930 


3025599 


! -433302 


•395734 


•330513 




20 


2-411714 


2-653297 
2-785962 


3-207135 


•414643 
•396787 


•376889-311804 




21 


■ ■ 
2-520241 


3*399563 


•358942/294155 




22 


2*633652 ! 2*92526Q 


3.603537 


•379701 


.341850 


•277505 




23 


2-752166 3-071523 


3.819749 


•363350 


•325571 


•261797 




24 


2-876013>225099 


4.048934 


•347703 


310068 


•246978 




25 


3-005434 


3 386354 
3-555672 


4,291870 


•332731 


305303 


•232998 




26 


3-140679 


4.549382 


•318402 


281241 


•219810 




27 


3-282009;3 733456 


4.822345 


•504691 


267848 


•207368 




28 


3-429699 3-920129 


5.111686 


•291571 


255094 


175630 




29 


3-584036 4-11 61 $5 


5.418387 


•279015 


•242946 


184556 




30 


3-7453 18'4-32 1942 


5.743491 


•267000 


231377 


17411o 




31 


3-913857J4-538039 


6.088 100 


•255552 


220359 


164255 




32 


408998 1;4-7 64941 


6.453386 


•244500 


209866 


154957 




3.3 


4-274030 5003188 


6.840589 


•233971 


199872 


146186 




o4 


4-466361 


5253347 


7.251025 


•223896 


190355 


137912 




35 


4-667347 


5-516015 


7.686086 


•214251 
•205028 


•181290 


130105 




36 


4*877378 


5 791810 


8.147252 


172057 


122741 




57 


,5-096860 


6 081406 1 8.636067 


•196299 


164436 


115793 




38 


5326219 


6385477J 9.154252 


•187750 


•156eo") 


109182 




39 


$565899 


6 704751? 9 703507 


•179659 


149148 


103002 




40 


5-816464 


7 039988 10285717 


•i7i9M-VAa^-vV»yivt^\ 



ho 



LQGARITHMICK 



Ta*l* III. Shotting the amount 
of$\ or £1 annuity for any num- 
ber of 'year *, from 1 to 40. 






Table IV^ Shewing tht 
present worth of $ 1 or £1 
annuity , for any number of 
y ear H y from 1 to 40. 



2 
3 
4 
5 

6 

7 

8 

9 

10 

11 
12 
13 
14 
15 

16 
17 
18 
19 
20 

21 
22 
23 
24 

25 



26 
27 

ZB 

29 

50 

31 
32 
33 
34 
35 



56 

>7 
38 
39 



1 000000 
2*045000 
3-137025 
4-278191 
5-470710 



6716892 

8-019152 

9-380014 

10-802114 

12-288200 



1 3.84 1179 
15.494032 
17-159913 
18.932109 
20-784054 



22-719337 
24741707 
26-855084 
29-063562 
31-371423 



33:783137 
36-303578 
38-937030 
41-689196 
44.565210 



47-570645 
50*711324 
53-993333 
57-423Q33 
61-00,7067 



64-752388 
68-666245 
72-756226 
77-030256 
81-496618 



1 -000000 
2-050000 
3-152505 
4-310125 
5-525631 



6-801913 

8- 142008 

9549109 

11026564 

12-577892 



14-206787 
15917126 
17-712985 
19-598632 
21-578563 



23-667492 
25-840366 
28-132385 
30-529003 
33-065954 



35-719252 
38-505214 
41-430475 
44-501999 
47-727099 



51*114454 

54-669126 

58-402583 

62-32271 

66-438847 



70-760790 
75-298829 
80-063771 
85066959 
90-320307 



£6-16396f 

9 1 04 1 344 

96-138206 

101-464424 



95-836323 
101-628139 
107-709546 
114095025 
120-799774, 



6 fir, cent 



1 000000 

2 O60000 
31836oo 
4-374916 
5 637093 



6-975318 

8-393837 

9-897467 

11-491315 

13-180794 



14-971642 
16-869940 
18-882132 
2 10 1 5064 
23-275968 



25-672527 
28-212879 
36-905652 
33-759991 
36-785590 



39-992725 
43-392289 
46-995826 
50-815576 
54-864510 



59-156381 
63705763 
68-528109 
73-639796 
79-058183 



84801674 
90.889775 
97-343161 
104183751 
111-434776 



119' 120863 
127-268114 
435-904201 
145*058453 
15476196\ 



^\h r. cf 5 fir. cl. 6 fir. ct. 



0-95694 
1-87267 
2-74896 
3-58752 
438997 



515789 
5-89270 
6.59589 
7-26879 
7-9 1272 



0-95238 
I 85941 
2-72325 
3-54595 
4-32948 

5-07569 
5-78637 
6-46321 
7-10782 
7-72x73 



8-52892- 8-30641 

9-11858? 8-86325 

9-68285= 9-39357 

10-22282;. 9-89864 

10-73954 10-37966 



11-2340M0-85777 
1 1-707 19 ! II'27407 
12-l5099|lI-68958 
12-59329 12-08532 
1300793 l2-4622i 



13-40793 I2-82I 15 
13-78442 1 3- 1 63oo 



14 14777 
14-49548 
I48282i 

15-14661 

15-45150 

15 74287 
1 6-02189 

16 28889 



16-54439 
16 78889 
1702286 
17-24676 
17-46I0I 

17-66604 
17-86224 
18 04999 
18-22965 
\8-40|58 



13-48807 
13-79864 
1409394 

14 37518 
l4-643o3 
1 4-898 1 3 
I*-x4lo7 
15-37245 



I55928I 
l5-8o268 
l6oo255 
16 i929o 
16-37419 



0-94339 
1.83339 
2-6730X 
3-465IO 
4-2x236 



4-91732 
5-58238 
6-2o976 
6-8ol69 
7-360O8 



7*88687 
838384 
8<85268 
9-29498 
9-71225 

IO 10589 
10-47726 

10 827fo 
IIl58ii 

1 1 46992 



11-7640? 
12-04X58 
l~-3o338 
l2-55o35 
12*78335 



i3oo3i6 
l3-2ro53 
13-40616 
l3-59o7 

13-76483 



13 92908 
I408398 
14-22917 
44 366 13 

14 49533 1 



16-54685 
167x129 
16-86789 
i7oi7o4 
l2.i59o9 



1461722 
14 7S211 
1 4 84048 1 
I4P4270J 
15-Q5913* 



ARITHMETICS 



141 



The annuity which 8 1, or £ 1 will purchase for any number. of 
years to come y from 1 to 40. 
• y"»»g» qeaaaaa « aaaa»o»tteaa«aaaa«a » B aaa ae e eaea» 
* * - **' Ai fler cent ' 5 fi er ccnt - I 6 fi*r cent. I TrTf 

^ 4 1 04500 105000 I 1-06000 1 • 



•53400 
•36377 
•27874 
•22779 

•19388 
•16970 
•15161 
•13757 
•12638 

•10967 
•10327 
•09782 
•093 1 1 

•08901 
•08542 
•08224 

•07941, 
'07688 

•07460 
•07254 
•07068 
'06899 
"06744 



•06602 
•06472 
•06352 
•06241 
•06139 

•06044 
•0595.6 
•05874 
•05798 
•05727 

•05660 
•05224 
•05540 
•0$485 
•05434 



53780 
•36721 
28201 
2309]T 

19702 
17282 
15473 
14069 
12950 



•12039, 
•11282 
•10645 
•10102 
•09624 



•09227 
'08870 
•08555 
•08274 
•08024 

«— -*-i--r i ».| 

•07800 
•07597 
•07414 
•07247 
•07095 

•06956 
•06829 
•06712 
•06604 
•06505 

•06413 
•06328 
•06249 
•06175 
«O6107 

•06043 
•05984 
•05928 
•05876 
•05?28 



•54544 
•37411 
•28859 
•23739 

•20336* 

•17913 

•16103 

•14702 

•13587 




•09895 


16 


•09544 


17 


•09235 


18 


•08962 


19 


•08718 


20 


•08500 


21 



•08303 
•08128 
•07968 
•07823? 



•07690 
•07570 
•07459 
•07358 
•07272 



07179 
<07100 
•07027 
•06959 
•06899 



•06839 
•06785 
•06735 
<Q6689 
•06646 



6 

7 

8 

9 

10 



11 
12 
13 
14 
15 



V 

\ 

K 

\ 

V 

s 

V 

V 
V 

I 

\ 



PRACTICAL ASTRONOMY. 



Containing a number of Astronomical Tables, and are an easy 
method of calculating the times of New and Full Moons, and 
Eclipses by them. 



Of Astronomical Tables and their Construction. 

IN constructing tables for computing, at any given instant, 
the places of the Sun, Moon, and Planets, the first step is to de» 
termine, from a series of accurate observations, the time in 
which those bodies describe a space of 360 degrees, or perform 
a complete revolution round the Sun, or primary Planet. 

When this important element is exactly ascertained, we can 
easily find, by simple proportion, the space which any Planet de- 
scribes in any number of years, months, days, hours, minutes, 
and seconds, upon the supposition that it moves uniformly, or 
describes equal spaces in equal time, in the circumference of a 
circle. 

But as it has been found from a long series of observations, 
that all the bodies of the solar system, move in eliptical orbits 
round the Sun; or their primary Planet, placed in one of the fo- 
ci, we must next determine the form of their orbits, or the na- 
ture of the ellipse.* which they describe. 

The diameters of the Sun and Moon therefore, subtend differ- 
ent angles at different times, as they aije nearer, or more remote 
from the observer's eye. This proves that the Sun; and Moon 
are constantly changipg their distances from the Earth,; and 
they are once at their greatest, and once at their least distance 
from it, in little more than a complete revolution. 

The gradual differences of these angles are not what they 
would be, if the luminaries moved in circular orbits, the Earth 



* An eiiipse is a curvilinear figure of an oblate oval form, 
having two centres called Foci, ortfoucuses: The Sun is in 
the focus of the Earths orbit, and the Earth is in or near that of 
the Moon's orbit. ' ' 



ARITHMETICS 1 43 

being supposed to be placed at some distance from the centre 
of the orbit, and the centre of the Earth to be in the lower focus 
of each orbit. 

The fartherest point of each orbit from the Earth's centre is 
called the Apogee, and the nearest point is called the Perigee* 
These points are diametrically opposite to each other. 

Astronomers divide each orbit into 12 equal parts called 
Signs ; each sign into 30 equal parts called Degrees ; each 
degree into 60 equal parts called Minuets ; each minuet into 
60 equal parts called Seconds. 

The distance of the Sun or Moon from any given point of its 
orbit, is reckoned in signs, degrees, minuets, and seconds. Here* 
in is meant the distance that the luminary has moved through 
from any given point : and not the space it is short of it in com* 
ing round again, though it be ever so little. 

The distance of the Sun or Moon from its apogee at any given 
time, is called its Mean Anomaly : therefore when the body is 
in its afiogccy its anomaly is 0, and in its ficrigee, it is 6 signs. 

The motion of the Sun and Moon are observed to be contin- 
ually accellerated from their apogee to their perigee, and afc 
gradually retarded from their perigee to their apogee ; moving 
with the greatest velocity when the anomaly is 0, and with the 
least, when the anomaly is 6 signs. 

When the luminary is in its apogee or its perigee, its place 
is the same as it would be, if its motion were equable in all parts, 
of its orbit. The supposed equable motions are called Mean ; 
the unequable motions are justly called the True. 

The mean place of the Sun or Moon is always forwarder than 
the true place,* while the luminary is moving from its apogee 
to its perigee ; and the true place is always forwarder than the 
mean, while the luminary is moving from its perigee to its apo- 
gee. In the former case the anomaly is always less than 6 signs; 
and in Jhe latter case, more. 

The Moon's orbit crosses the ecliptick in two opposite points, 
which are called her Nodes ; and the time in which she revolves 
from the Sun to the Sun again, (or from change to change) is 
called a Lunation, and would always consist of 29 days, lfc 
hours, 44 minuets, 3 seconds, 2 thirds, and 58 fourths, if the mo- 
tions of the Sun and Moon were always equable. Hence, 1 2 mean 
lunations contain 354 days, 8 hours, 48 minuets, 36 seconds, 3$ 
thirds, and 40 fourths, which is 10 days, 21 hours, 11 minuets* 
23 seconds, 24 thirds, and 20 fourths, less than the length of a 
common Julian Year, consisting of 365 days 6 hours; and IS 



*The point of the ecliptick in which the Sun or Moon isittL 
any moment of time is called the place of the Sua or Moonai' 
that time, 



144 LOGARITHMICK 

mean lunations contain 383 days, 21 hours, 32 minuets, 59 sec- 
onds, 38 thirds, and 38 fourths, which exceeds the length of a 
common Julian Year, by 18 days, 15 hours, 32 minuets, 39 
seconds, 38 thirds, and 38 fourths. 

The mean time of New Moon being found for any given year 
and month, as suppose for March 1850, New Style, if this mean 
New Moon happens later than the 1 1th day of March, then 12 
mean lunations, added to the time of this mean New Moon, will 
give the time of the mean New Moon in March 1851, after aba- 
ting 365 days. But when the mean New Moon happens to be 
before the 1 1th of March, we must add 13 mean lunations, in 
order to have the time of mean New Moon in March the year 
following ; observing always to subtract 365 days in common 
years, and 366 days in leap-years, from the sum of this addition. 

Thus, A. D. 1850, JWw Style, the time of mean New Moon 
in March-, was the 12th day, at 22 hours and 1 1 seconds past the 
noon of that day (viz. at 1 1 seconds past X in the morning of the 
J 3th day, according to common reckoning.) To this we must 
add 12 mean luxations, or 354 days, 8 hours, 48 minuets, 36 
seconds, 35 thirds, and 40 fourths, and the sum will be 367 days, 
6 hours, 48 minuets, 47 seconds, 35 thirds and 40 fourths ; from 
which subtract 365 days, because the year 1851 is a common 
year, and there will remain 2 days, 6 hours, 48 minuets? 47 
seconds', 35 thirds and 40 fourths, for the time of mean New- 
Moon in March, A. D. 1851. 

Now to find the mean time of New Moon in March A. D. 
1852, we must add 13 mean lunations to the mean time of New 
Moon in the next precfecding year, (because it happened before 
the 1 1th day)a'nd the sum will be 386 days, 4 hours 21 minuets 27 
seconds 13 thirds and 18 fourths ; from which subtract 366 days, 
because the year 1852 is a leap-year, and there will remain 20 
days 4 hours 21 minuets 27 seconds 13 thirds and 18 fourths, to 
be set down for the time of mean New Moon, in March, A; D.. 
1852 

In this manner was the first two of the fallowing tables con- 
structed to seconds, thirds, fourths ; and then written out to 
the nearest second. The reason why Astronomers choose to be- 
gin the year with March, is to avoid the inconvenience of ad- 
ding a day to the tabular time in leap-years after February, o* 
subtracting a day therefrom in January and February in those 
years ; to which all tables of this kind are subject, which be- 
gin tho year with January, in calculating the times of New or 
Full Moons. 

The mean anomalies of the Sun and Moon, and the Sun's 
mean distance from the ascending node of the Moon's orbit, are 
set down in Table III, from one to 1 3 mean lunations. 

The numbers, for 12 lunations, being added to the radical 
anomalies of the Sun and Moon, and to the Sun's mean distance 



ARITHMETICS U* 

from the Moon's ascending node, at the mean time of New 
Moon in March 1850, (Table II.) will give their mean anoma- 
lies, and the Sun's mean distance from the node, at the time of 
New Moon in March 1851 ; and being added for 13 lunations 
to those for 1851, will give them for the time of mean New 
Moon in March 1852. And so on as far^as you please to con- 
tinue the table, (which is here carried on from 1752, to the year 
1900,) always rejecting 12 signs when their sum exceeds J 2, 
and setting down the remainder as the proper quantity. 

If the number of years belonging to A. D. 1700 (in Table I.) 
be subtracted from those belonging to 1800, we shall have their 
whole differences in 100 complete Julian years ; which accord- 
ingly we find to be 4 days 8 hours 10 minuets 52 seconds IS 
thirds 40 fourths, with respect to the time of mean New Moon. 
These being added together 60 times, (always taking care to 
throw off a whole lunation when the days exceed 29*) making 
up 60 centuries, or 6000 years, as in Table VI. whicu was car- 
ried on to seconds, thirds, ami fourths ; and then written out to 
the nearest second. In the same manner were the respective 
anomalies and the Sun's distance from the node found, for these 
centural years ;and then (for want of room) written out only to 
the nearest minuet, which is sufficient in whole centuries. By 
means of these two tables, we may find the time of any mean 
New Moon in March, together with the anomalies of the Sun 
and Moon, and the Sun's distance from the node, at these times, 
within the limits of 6000 years, either before or after any given 
year in the 18th: century ; and the mean time of any. New or 
Full Moon in any month of the year after March, by means of . 
the third and fourth tables, within the same limits, as shown in 
the precepts for calculation. 

Thus it would be a very easy matter to calculate the time of 
any New or Full Moon, if the Sun and Moon moved equably ia 
all parts of their orbits. But we have already observed that 
their places are never the same as they would be by equable 
motions, except when their mean anomalies are cither 0, or 6 
signs ; and that their mean places are always forwarder than 
than their true places, while their anomalies are less than 6 
signs ; and their true places are forwarder than the mean, while 
the anomaly is more. 

Hence it is evident, that while the Sun's anonaly is less than 
6 signs, the Moon will overtake him, or be opposite to him, 
sooner than she could if his motion were equable ; and later 
while his anomaly is mora than 6 signs. The greatest differ- 
ence that can possibly happen between the mean arid true time 
of New or Full Moon, on account of the inequality of the Sun's 
motion, is 3 hours 48 minuets 28 seconds ; and this is when 
the Sun's anomaly is either 3 signs 1 degree, or 8 signs 29 de* 
grees ; sooner in the first cas-^, and later in the last* lit. all 



I 



146 LOGARITHMICK 

other signs and degrees of anomaly, the difference is gradually 
lets, and vanishes when the anomaly is either 0, or 6 signs. 

The Sun is in his apogee on the 30th. of June, and in his 
perigee on the 30th. of December, in the present age ; so that 
he is nearer the earth in our winter than in our summer. The 
proportional difference of the Sun's apparent diameter at these 
limes, is as 983 to 1017. 

The Moon's orbit is dilated in winter, and contracted in sum- 
mer. The greatest difference is found to be 22 minuets 39 
seconds ; the lunations increasing gradually in length while 
the Sun is moving from his apogee, and decreasing in length 
while he is moving from his perigee to his apogee. On this 
account the Moon will be later in coming to her conjunction 
with the Sun, or being in oppositions him,* from December till 
June, and sooner from June to December, than if her orbit had 
continued of the same size all the year round. 

As both these differences depend on the Sun's anomaly, they 
may be fitly put together into one table and called The annual, 
orjirat equation for reducing the mean to the true syzygy.\ (See 
Table VII.) 

This equational difference is to be subtracted from the time 
of mean New or Full Moon when the Sun's anomaly is less 
than 6 signs, and added when the anomaly is more. 

At the greatest, it is 4 hours 10 minuets 57 seconds, Viz. 3 
hours 48 minuets 28 seconds, on account of the Sun's unequal 
motion, and 22 minuets 29 seconus, on account of the dilation 
of the Moon's orbit. 

This compound equation would be snmcient for reducing the 
moan time of New or Full Moon to the true time, if the Moon's 
orbit were of a circular form, and her motion exactly equable in 
it. But the Moon's orbit is more elliptical than the Sun's, and 
her motion in it is so much the more unequal. The difference 
is so great, that she is sometimes in conjunction with the Sun, 
or in opposition to him, sooner by 9 hours 47 minuets 54 sec- 
onds, than she would be if her motion were equable ; and at 
other times as much later. The former happens when her 
mean anomaly is 9 signs 4 degrees, and the latter when it is 2 
signs 26 degrees. See Table IX. 

At different distances of the Sun from the Moon's apogee, the 
figure of the Moon's orbit becomes different. 

It is longest of all, or most excentrick, when the Sun is in 



* The term conjunction, when it respects the relation of the. 
Moon to the Sun, signifies New Moon, or Change ; and oppo- 
sition is used to signify the place of the Moon at her full. 

tThe word syzygy siguifies both the conjunction and opposi- 
tion of the Sun and Moon. 



ARITHMETIC!*. iff 

the same sign and degree either with the Moon's apogee or 
perigee ; shortest of all, or least exccntrick, when the Sun's 
distance from the Moon's apogee is either 3 signs or 9 signs ; 
and at a mean state when the distance is either 1 sign 1 5 degrees, 
4 signs 15 degrees, 7 signs 15 degrees, or 10 signs 15 degrees. 
When the Moon's orbit is at its greatest excentricity, her apo- 
geal distance from the Earth's centre is to her perigeal dis- 
tance from it, as 1067 is to 933 > when least excentiick, as 1043 
is to 457, and when at the mean state, as 1055 is to 945. 

But the Sun's distance from the Moon's apogee is equal to 
the quantity of the Moon's mean anomaly at the time of New 
Moon, and by the addition of 6 signs, it becomes equal in quan- 
tity to. the Moon's mean anomaly at the time of Full Moon. 
Therefore, a table may be constructed so as to answer all the 
various irregualities depending on the different excenlricities 
of the Moon's orbit in the syzygies ; and called The second 
equation for reducing the mean to the true syzygy, (See Table 
IX.) and the Moon's anomaly, when equated by Table^VIH. 
may be made the proper argument for taking out this second 
equation of time, which must be added to the former equated 
time, when the Moon's anomaly is less than 6 signs, and sub- 
tracted when the anomaly is more. 

There are several other inequalities in the Moon's motion, 
which sometimes bring the true syzygy a litle sooner, and at 
other times keep it back aiittle later than it would otherwise 
be ; but they are so small, that they may be all omitted except 
two ; the former of which (See Table X.) depends on the differ- 
ence between the anomalies of the Sun and Moon in the syzygies 
«nd, the latter, (see Table XI.) depends on the Sun's distance 
from the Moon's nodes at these times. The greatest difference 
arising from the former, is 4 minuets 58 seconds ; and from 
the latter, 1 minuet 34 seconds. 

Besides the tables already mentioned, there are various others 
annexed in the following, to facilitate the labour of astronomi- 
cal calculations* and will be treated of in their proper place. 



+ •*■ ■& 

* •$ * i 
■%■ •$• -£ 

•*• ■*• 
■*■ * . 

.St. 



TABLES 



ARITHMET1CK. 



149 



TABLES 

FOR CALCULATING THE TRUE TIME OF NEW 

AND FULL MQOA'S AND ECLIPSES. 
TABLE I. The mean Time of New Mjoh in March, Old Style, 
with the mean Anomalies of the Sun and Moon, and the Sun 9 * 
Mean Distance from the Moon's Aacinding Node, from A. D 
1700 to A. D. 800 inclusive 



O 

S3* 

1700 
1701 
1702 
1703 
1704 



Mean iNcw 

Moon 
in March. 



D. H. M. S. 



8 16 11 

27 13.44 

16 22 32 

6 7 2i 

24 4 53 



dun s mean 
Anomaly. 



O f tf 



Moon's mcuij Sun's mean 



Anomaly 







1705 
1706 
1707 
1708 
1709 



1710 
1711 
1712 
1713 
1714 



13 13 42 

2 22 31 

31 20 3 

10 4 52 

29 2 25 



258 

59 

418 

188 

il'L 

348 
118 
509 
278 
79 



lisiance from 
the Node. 



19 58 48 1 22 30 3? 

8 20 59 28 

27 36 5l'll 7 

16 52 43 9 17 

5 14 54' 8 23 



55 47 
43 52 



20 57 



42 7 
8 
8 
9 



14 31 7 

23 14 8 

1 16 55 

9 19 42 

18 2 43 



24 30 47i 7 3 9 2 

13 46 39 5 12 57 7 

2 8 50 4 18 34 13 

21 24 48 2 28 22 18 

9 46 54' 2 3 59 24 



18 11 13 

7 20 2 

25 17 34 

15 2 23 

4 11 12 



1715 
1716 
1717 



1718 19 
1719 



438 
208 
599 
36 8 
13 8 



23 

11 

1 



8 44 
17 33 



2 
23 



22 

54 



8 43 



1720 27 
172116 

1722 5 

1723 24 

1724 13 



1725 2 15 

1726 21 12 
10 21 



1727 
1728 
1629 



6 
15 
23 
fil 

6 



16 
4 



ly 
38 



29 2 471 

18 18 3910 

6 40 51 9 

25 56 431 8 

15 12 35 6 



3 34 47 

22 50 39 

12 6 32 

28 44 



13 47 30 

23 35 36 

29 12 42 

9 47 

18 48 52 



26 5 30 

4 8 17 

12 51 18 

20 54 5 

29 37 6 



8 19 44 3711 



1 
1 
2 
3 

3_ 

24 25 571 4 
4 14 2 4 
14 2 8 5 
19 39 13, 6 
29 27 18 6 



7 39 54 

15 42 41 

14 25 43 

2 28 30 

10 31 17 



19 14 18 

27 17 5 

5 19 52 

14 2 54 

22 5 41 



53 14 
25 54 
14 31 



28 18 
18 3 



1730 7 

1731 26 
173V 14 

1733 4 

1734 23 



3 78 
35 47 9 
24 23 8 
57 * 3 9 
45 40 8 

8" 



8 
27 
16 

5 

24 



6 
22 
38 


16 



4911 



5 4 241 8 

14 52 29 8 

24 40 34 8 

17 40 9 

10 5 45 10 



48 43 

8 51 29 

16 54 16 

25 37 18 

3 40 5 



13 32 291 3 19 

1 54 41 2 25 

21 10 34 1 5 

9 52 46 10 

28 48 3910 20 



5010 
5611 

1 11 



34 16 

6 56 

55 33 
44 

16 4S 



18 4 31 9 

6 26 42 8 6 

25 42 34 6 15 

U 58 26 4 25 

3 20 38 4 1 



17 
23 
28 
S3 
39 



11 42 52 

20 25 54 

28 28 41 

7 11 42 

15 14 29 



23 17 16 

2 17 

10 3 4 

18 5 51 

26 48 51 



17*o|12 10 3 25 8 22 36 30 2 11 10 44 

175 C 13 54 2 8 11 52 22 20 58 49 

1737'. 19 16 26 42^9 14 3411 26 35 55 

1738 9 1 15 18 8 19 30 2610 6 24 

1739127 22 47 5819 7 52 38! 9 12 1 6 

20 



5 4 51 40 

5 12 5i 27 

6 21 37 29 
6 29 40 16 
8 8 23 18 



150 



LOGARITHMIC* 





TABLL 


1. 


continued. Old Slytt. 


< 


Mean New 


b 


un b 


mean 


Moon's mean 


Sun's mean 


o 




Moon 




Anomaly. 


Anomaly. 


distance from 


O 

1 E 


li 
D. 


i March. 








n 




the Node. 


H. M. S 


s 





/ 


s ' f '\ s ' '' 


1740 


16 


7 36 34 


3 


27 


8 


30 


7 21 49 Hi 8 16 26 5 


1741 


5 


16 25 11 


8 


16 


24 


22 


6 1 37 16 8 24 28 52 


1742 


24 


13 57 52 9 


4 


46 


34 


5 7 14 2210 3 11 54 


1743 


13 


22 46 27,8 


24 


2 


27 


3 17 2 27,10 1 14 41 


1744 
1745 


2 


7 35 43 


13 


18 


20 


1 26 50 32 


10 19 17 28 

11 28 30 


21 


5 7 449 


1 


40 


32 


1 2 27 38 


1746 


10 


13 56 2t)8 


20 


56 


24 


11 12 15 43 


6 3 17 


1747 


29 


11 29 09 


9 


18 


3610 17 52 49 


1 14 46 19 


1748 


17 


20 17 368 


28 


34 


28 


8 27 40 54 


1 22 49 5 


1749 
1750 


7 


5 6 13-8 


17 


50 


20 


7 7 28 59 


[ 2 ♦ 51 52 


26 


2 38 53,9 


6 


12 


32 


6 13 6 5 


3 9 34 53 


1751 


15 


11 27 29|8 


25 


28 


24 


4 23 54 10 3 17 37 40 


1752 


3 


20 16 6 3 


14 


44 


16 


3 2 42 15 


3 35 40 27 


1753 


22 


17 48 45 9 


O 


6 


28 


2 6 19 21 


S 4 23 28 


1754 
1755 


12 


2 37 228 


22 


22 


20 


11 7 26 


5 12 26 15 


1 


11 25 598 


11 


38 


12 


10 27 55 31 


5 20 29 2 


1756 


19 


8 58 389 








24 


10 3 32 37 


6 29 12 3 


1757 


8 


17 47 15 8 


19 


16 


16 


8 13 20 42 


7 7 14 50 


1758 


27 


15 19 549 


7 


38 


88 


7 28 57 4fc 


8 15 57 52 


1759 
1760 


17 


8 318 


26 


54 


20 


5 28 45 54 


8 24 39 


5 


8 57 8 8 


16 


10 


12 


4 « 34 6 


9 2 3 26 


1761 


24 


6 29 479 


4 


32 


24 


3 14 11 6 


10 10 46 27 


1762 


13 


15 18 248 


23 


48 


16 


1 23 59 11 


10 18 49 14 


1763 


3 


7 1 


8 


13 


4 


8 


3 47 16 


10 26 52 1 


1764 
1765 


20 


21 39 40 


9 


1 


26 
42 


20 
13 


11 9 24 21 


5 35 2 


10 


6 28 17 


8 


20 


9 19 12 26 


13 37 49 


1766 


29 


4 56 


9 


9 


4 


20 


8 24 49 35^ 


1 22 20 51 


1767 


18 


12 49 33 


« 


28 


20 


17 


7 4 37 37 


2 23 38 


1768 


6 


21 38 10 


8 


17 


36 


9 


5 14 25 42 


2 8 26 25 


1769 
1770 


25 


19 . 10 40 


9 


5 


58 


21 


4 20 2 48 


1 3 17 9 27 


15 


3 59 26 


8 


25 


14 


13 


2 29 50 53 


3 25 12 14 


1771 


4 


12 48 2 


8 


14 


30 


5 


1 9 38 58 


4 3 15 1 


1772 


22 


10 20 43 


9 


2 


52 


17 


15 16 4 


5 11 58 3 


1773 


11 


19 9 19 


8 


22 


8 


9 


10 25 4 9 


5 20 50 


1774 


1 


8 57 55 


8 


.11 


24 


1 


9 4 52 14 


5 28 3 37 


1775 


20 


1 30 25 


8 


29 


46 


13 


8 10 29 20 


7 6 49 38 


1776 


* 


10 19 12 


8 


19 


2 


5 


6 20 17 25 


7 14 49 25 


1777 


27 


7 51 51 


9 


7 


24 


17 


5 25 54 31 


8 23 32 26 


1778 


16 


16 40 28 


8 


26 


40 


9 


4 5 42 86 


9 1 35 13 


177Q 


6 


I 29 4 


8 


15 


56 


1 


2 15 30 41 


9 9 38 


1780 


23 


23 1 44 


9 


4 


18 


13 


1 21 7 47 


10 18 21 1 


1781 


13 


7 50 21 


8 


23 


34 


5 


55 52 


10 26 23 48 


1782 


2 


16 38 57* 


8. 


12 


49 


58 


10 10 43 57 


11 4 26 35 


1783 


21 


14 11 37 


9 


1 


12 


10 


9 16 21 3 


13 9 36 


1784 
1785 


9 
28 


23 13 


3 


20 


28 


3 


7 26 9 8 


21 12 23 


U0 32 56[9 


9 


50 


15 


7 1 46 14 


1 29 55 25 


"' 17861 


18 


5 21 30'8 


28 


6 


71 5 11 34 19 


2 7 58 12 



ARITHMET1CK. 



151 



1 






TABLE 


1. concluded. 


Old Styl 


c. 






*: 


M 


tun 


New 


Sun's, mean 


Moon' 


s meai. 


Sun's .mean 


o 




Moon 




Anomaly 




' Anomaly. 


'distance from 


O 


in 


March. 








s 


O 






the > 
s 


J ode. 


D. 


H. 


M. S 


s 


'. 


// 


/ 


// 


/ a 


1787 


7 


14 


10 6 8 


17 21 


59 


3 


21 


22 


24 


2 


16 


59 


1788 


25 


11 


42 46 9 


5 44 


11 


2 


26 


59 


30 


3 


24 


44 1 


1789 


14 


20 


31 238 


25 


3 


1 


6 


47 


35 


4 


2 


46 48 


1790 


4 


5 


19 59 8 


14 15 


5511 


16 


35 


40 


4 


10 


49 35 


1791 


23 


2 


52 39 


9 
8 


2^ 38 
21 53 


7 
59 


10 


22 


12 


46 


5 


19 


32 37 


1792 


11 


11 


41 15. 


9 


2 


U 


52 


5 


27 


35 24 


1793 


30 


9 


13 559 


10 16 


11 8 


7 


37 


58 


7 


6 


18 26 


1794 


19 


18 


2 32'8 


29<> 32 


3 6 


17 


26 


4 


7 


14 


21 13 


1795 


9 


2 


51 88 


18 47 


5S, 4 


27 


14 


9 


7 


22 


24 


1796 27 

17971 16 

1798) -5 




9 


23 48 

12 24. 


9 


7 10 


71 4 


2 


51 


14 


9 


1 


7 1 


8 


26 tS 


59 2 


12 


3* 


19 


9 


9 


9 48 


18 


1 18 


15 41 


51' 


22 


27 


25 


9 


17 


12 35 


1799,24 


15 


23 41 9 


4 4 


311 


28 


4 


31 


10 


25 


55 37 


1800] 


13 





22 17 


8 


23 19 


5510 


7 


52 


36 


11 


3 


58 22 



TAULk 11. Mean New AJoon, <Jfc.iu iviaixh, *viu; o; 


yw, ji-om 








A. D. 1752 to A. 


D. 19 r 0. * 








^ 


Meat 


New 


Sun's 


mean 


Moon's meai. 


■'JJUII'ji 


mean 




P 




Moon 


Anomaly. 


. 


Anomaly. 


distant 


:e from 




O 


in M 


arch. 














the Node. 




-i 




























CD 


P. 


H. 


M. S 


s 





/ a 


s 


' 


ii 


s 


O 


i a 




1752 


14 


20 


16 6 


8 


14 


44 16 


3 


2 42 


15 


3 


25 


4U 27 




1753 


4 


5 


4 42 


8 


4 


8 


1 


12 30 


20 


4 


3 


43 14 




1754 


23 


2 


37 22 


8 


22 


22 20 





18 7 


26 


5 


12 


26 15 




1755 


12 


11 


25 59 


8 


11 


38 12 


10 


27 55 


31 


5 


20 


29 2 




1756 


30 


8 


58 38 


9 





24 


10 


3 32 


37 


6 


29 


12 3 




1757 


19 


17 


47 IS 


8 


19 


16 16 


8 


13 20 


42 


7 


7 


14 50 




1758 


9 


2 


35 51 


8 


8 


32 8 


9 


23 8 


47 


7 


15 


17 38 




1759 


28 





8 31 


8 


26 


54 20 


5 


28 45 


54 


8 


24 


39 




1760 


16 


8 


57 8 


8 


16 


10 12 


4 


8 34 





9 


2 


3 26 




1761 
1762 


5 


17 


45 44 


8 


5 


26> 4 


2 


18 22 


5 


9 


10 


6 13 




24 


15 


18 24 


8 


23 


48 16 


1 


23 59 


lljlO 
16'10 


18 


49 14 




1763 


14 





7 1 


8 


13 


4 8 





3 47 


26 


52 1 




1764 


2 


8 


55 36 


8 


2 


20. 


10 


13 35 


21 


11 


4 


54 48 t 




1765 


21 


6 


28 17 


8 


20 


42 13 


9 


19 12 


26 





13 


37 49 




1766 


10 


15 


16 53 


8 


9 


58 5\ 7 


29 


31 





21 


40 37 




1767 


29 


12 


49 33 


8 


28 


20 17 


7 


4 37 


3/ 


2 





~3 38 




1768 


17 


21 


38 9 


8 


17 


36 9 


5 


14 25 


42 


2 


8 


26 25 




1769 


7 


6 


26 46 


8 


6 


52 1 


3 


24 13 


47 


2 


16 


29 13 




1770 26 


3 


59 26 


8 


25 


14 13 2 


29 50 


53 


3 


25 


12 14 




1771 Il5 


12 


48 2 


8 


14 


30 5' 1 


9 38 


58 4 


3 


15 1 




17721 3 
117SI22 


21 


36 39 


8 


3 


45 57U 


19 27 


3 4 


11 


17 48 




19 


9 19 


8 


ae 


8 9 10 


25 4 


9 5 


20 


50 




177412 


3 


57 55 


8 


n 


24 1 9 


4 52 


14 5 


28 


3 37 




17751 1 


12 


46 31 


8 





39 53 7 


14 40 


19 6 


6 


fc *i\ \ 


1776 


19 


10 


19 12 


8 


19 


2 6 6 


20 17 


2,5 7 


\4i 


AS) ^5 





452 



LOGARITHMIC* 



* 


fAHLE 11. continued. 


JVew Style. 




^ 


Mean iNcvv 


buu'fc 


mean 


Moon's meant Sun's mean 


c 


Moon. 


Anomaly. 


Anomaly, (distance from 


O 

Or 


in March. 










the Node. 


D H. 


M. S. 


s 


f rf 


s 





' h 


s O 


/ tr 


1777 


8 19 


7 48 


8 8 


17 5f\ 5 





5 o0 


7 22 


52 12 


1778 


27 16 


40 28 


8 26 


40 9 4 


5 


42 36 


9 1 


35 13 


1379 


17 1 


29 4 


8 15 


56 1 2 


15 


30 41 


9 9 


38 


1780 


5 10 


15 3 


8 5 


5 54 


25 


22 58 


9 17 


36 12 


1781 


24 7 


47 40 


8 23 


28 4] 


1 


9 


10 26 


19 14 


1782 


13 16 


36 14 


8 12 


43 55*10 


10 


48 18 


11 4 


22 1 


1783 


3 1 


24 48 


8 1 


59 47 


8 


20 


36 28 


11 12 


24 49 


1784 


20 22 


37 25 


8 20 


21 57 


7 


26 


13 39 


21 


7 50 


1785 


10 7 


45 59 


8 9 


37 48 


6 


6 


1 49 


29 


10 38 


1786 


29 5 


18 36 


8 27 


59 58 


5 


11 


38 59 


2 7 


53 39 


1787 


18 14 


7 10 


8 17 


15 50 


3 


21 


27 9 


2 15 


56 26 


1788 


6 22 


55 45 


8 6 


31 40 


2 


1 


15 19 


2 23 


59 14 


1789 


25 20 


28 22 


8 24 


53 51 


1 


6 


52 30 


4 2 


42 15 


1790 


15 5 


16 56 


8 14 


9 42 


11 


16 


40 39 


4 10 


45 3 


1791 


4 14 


5 30 


8 3 


25 33 


9 


26 


28 49 


4 18 


47 50 


1792 


22 11 


38 7 


8 21 


47 44 


9 


2 


6 


5 27 


30 52 


1793 


11 20 


26 41 


8 11 


3 35 


7 


11 


54 10 


6 5 


33 39 


1294 


30 17 


5*. 18 


8 29 


25 45 


6 


17 


31 20 


7 14 


16 41 


1795 


20. 2 


47 53 


8 18 


41 36 


4 


27 


19 30 


7 22 


19 28 


1796 


8 11 


36 27 


8 7 


57 28 


3 


7 


7 40 


8 

9 9 


22 16 
5 17 


1797 


27 9 


9 4] 8 26 


19 38 


2 


12 


44 51 


1798 


16 17 


57 38 8 15 


35 29 





22 


33 


9 17 


8 5 


1799 


6 2 


46 12 8 4 


51 20 


11 


2 


21 10 


6 25 


10 52 


1800 


25 


18 49 8 23 


13 30 


10 


7 


58 21 


11 3 


53 54 


1801 


14 9 


7 23! 8 12 


29 22 


8 


17 


46 31 


11 11 


56 41 


1802 


3 17 


55 5b 


8 1 


45 13 


6 


27 


34 41 


11 19 


59 29 


1803 


22 15 


28 35 


8 20 


7 25 


6 


3 


11 51 


28 


42 30 


1804 


11 


17 fi 


8 9 


23 14 


4 


13 


1 


1 6 


45 18 


1405 


9 


5 43 


8 28 


39 5 


2 


22 


48 11 


1 14 


48 $ 


1806 
1807 


19 6 


38 20 


8 17 


1 16 


1 


28 


25 21 


2 23 

3 1 


31 7 
33 54 


b 15 


26 55-, 8 6 


17 7 





8 


13 31 


1808 


26 12 


59 311 8 24 


39 17 


11 


13 


50 42 


4 10 


16 56 


1809 


15 21 


48 5\ 8 13 


55 8 


9 


23 


38 52 


4 18 


19 43 


1810 


5 6 


36 40' 8 3 


10 59 


8 


3 


27 2 


4 26 


22 31 


1811 


24 4 


9 17| 8 21 


33 10 


7 


9 


4 12 


6 5 


5 32 


1812 


12 12 


57 51 8 10 


49 1 


5 


18 


52 22 


6 13 


8 20 


1813 


1 21 


46 25 8 


4 52 


3 


28 


40 32 


6 21 


11 7 


1814 


20 19 


19 2; 8 18 


27 3 


3 


4 


17 43 


7 29 


54 9 


1815 


10 4 


7 37 8 7 


42 54 


1 


14 


5 52 


8 7 


56 56 


1810 


2» 1 


40 14| 8 26 


5 4 





19 


43 3 


9 16 


39 58 


1317 17 10 


28 48 8 15 


20 55 


10 


29 


31 13 


9 24 


42 45 


1818 


6 19 


17 22 8 4 


36 46 


9 


9 


19 23 


10 2 


45 33 


1819 


25 16 


51 14| 8 23 


5 44 


8 


14 


41 17 


11 11 


29 55 


182J 


,14 1 


38 33 1 8 12 


14 48 


6 


24 


44 43 


11 19 


31 22 


18211 3 10 


27 7 8 1 


30 39 


5 


4 


32 53jll 27 


34 9 


1822|22 7 


49 45! 8 19 


52 50 


4 


10 


10 34| 1 6 


17 10 


1823 


111 16 


48 19 


I 8 9 


8 41 


2 


19 


58 13 


1 14 


19 58 



ARITHMETIC*. 



153 



TABLE 11. continued. JYcv> Style. 


o 

o 

sr 

CO 


Mean New 

Moon 
in March. 


Sun's Mean 
Anomaly. 


Moon's meai, 
Anomaly. 


Sun's mean 

distance from 

the Node. 


D. H. M. S 


s 


' " 


so'" 


s O ' " 


1824 
1825 
1826 
1827 
1828 


29 14 20 56 
18 23 9 SO 
8 7 59 4 
27 5 30 41 
15 14 19 15 


8 
8 
8 
8 
8 


27 30 51 
16 46 42 
6 2 33 
24 24 44 
13 40 35 


1 25 35 24 
5 33 34 
10 45 11 44 
9 20 48 54 
8 37 4 


2 23 2 59 

3 1 5 47 
3, 9 8 34 

4 17 51 36 
4 25 54 23 


18291 
1830: 
1831 
1832 
1833 

1834 
1835 
1836 
1837 
1838 


4 23 7 50 
23 20 40 27 
13 5 29 1 

1 14 17 35 
20 11 50 12 


8 
8 
8 
8 
8 


2 56 26 
21 18 36 
10 34 27 
29 50 19 
18 12 29 


6 10 25 15 
5 16 2 55 
3 25 50 34 
2 5 38 44 
1 11 15 55 


5 3 7 11 

6 12 40 12 
6 20 3 
6 28 45 47 
8 7 28 49' 


9 20 38 46 
28 18 11 23 
17 2 59 58 

6 11 48 32 
25 9 21 9 


8 
8 
8 
8 
8 


7 28 20 
25 50 31 
15 6 22 

4 22 13 
22 44 23 


11 21 4 4 

10 26 41 15 

9 6 29 25 

7 16 17 35 

6 21 54 46 


8 15 31 36 

9 24 14 38 
10 2 17 25 
t0 10 20 13 
U 19 3 14 


1839 
1840 
1841 
1842 
1843 


14 18 9 43 
3 2 58 17 
22 30 54 
11 9 19 28 
30 6 52 6 


8 
8 
8 
8 
8 


12 14 
1 16 5 

19 38 16 
8 54 7 

27 16 17 


5 1 42 55 
3 11 31 5 
2 17 8 16 
26 56 26 
2 33 36 


11 27 6 2 

5 8 49 

1 13 51 51 
1 21 54 38 
3 37 40 


1844 
1845 
1846 
1847 
1848 


18 15 40 40 

8 29 14 

26 22 1 51 

16 6 50 25 

4 15 38 59 


8 
8 
8 

|8 


16 32 8 

5 47 60 

24 10 10 

13 26 1 

• 2 41 5 


10 12 21 46 
8 22 9 56 
7 27 47 7 
6 7 35 16 
4 '17 23 26 


3 8 40 27 

3 16 43 15 

4 25 26 16 

5 3 29 4 
5 11 31 51 


1849 
1850 
1851 
1852 
1853 


23 13 11 36 
12 22 11 

2 6 49 48 
20 4 21 27 

9 13 10 4 


8 
8 
7 
8 
8 


21 4 3 
10 19 54 
29 35 46 
17 57 58 
7 13 5C 


3 23 -37 

2 2 48 47 

1 12 36 52 

'11 18 13 58 

I 9 28 2 3 


6 20 14 53 

6 28 17 40 

7 6 20 27 

8 15 3 28 
8 23 6 15 


1854 
1855 
1856 
1857 
1858 


28 10 42 43 
17 19 31 20 
► 5 4 19 57 
'24 1 52 36 
113 10 41 13 


i 8 

8 
8 
8 
8 


25 36 2 
14 51 54 
A 7 At 
22 29 5i 
11 43 5C 


9 3 39 9 
t 7 13 27 14 
> 5 23 15 19 
\ 4 28 52 25 
V 3 8 40 30 


10 1 49 16 
10 9 52 3 

10 17 54 50 

11 26 37 51 • 
4 40 38 


18591 2 19 29 49 
186020 17 2 29 
1861J10 1 51 6 
1862:28 22 23 45 


8 
8 
8 
8 


1 1 42| 1 18 28 35 
19 23 54' 24 5 41 

8 39 4611 3 53 4( 
27 1 5810 9 30 55 


12 43 25. 

1 21 46 26 
1 29 49 13 
3 8 32 14 


1863.17 7 12 22 
1864| 5 16 59 
1865 24 13 33 38 
186613 22 22 15 


8 

8 

1 8 

8 


16 17 50 8 19 18 57 

6 33 24' 6 29 7 S 

24 55 54' 6 4 14 I 

14 11 46 4 14 2 i: 


3 16 35 1 

3 24 37 48 
5 5 3 20 49 
* 5 11. 23 36 


1867j 3 7 10 51 
186821 4 43 31 
1869.10 13 31 8 
1870*28 11 4 47 


8 

8 

t 8 

' 8 


3 27 38. 2 23 50 1* 
21 49 50 1 29 27' 24 
11 5 4210 9 15 2* 
29 27 5411 14 52 3. 


J 5 19 26 23 
I 6 28 9 24 
) 7 6 12 11 
5 8 14 55 12 



154 



LOGARITHMIC* 





TAULk 


. 11 


. concluded, JS'ew 


Style. 




< 


Mean New 


Sun's 


Mean 


Moon 9 


a meaii 


Sun's mean 


Q 


Moon 




Anomaly. 


Anomaly. 


distance from 


o 
**• 


in March. 
















tl 
s 


ie Node. 


D. H. 


M. 


S. 


s 





r 


tf 


s O 


/ 


fi 


' " 


18/1 


17 12 


53 


24 


8 


18 


43 


46 


9 24 


40 


40 


8 


22 57 59 


1872 


5 4 


42 





8 


7 


59 


38 


8 4 


28 


45 


9 


1 46 


1873 


24 2 


15 


40 


8 


26 


21 


50 


7 10 


5 


5110 


9 43 47 


1874 


13 11 


4 


17 


8 


15 


37 


42 


5 19 


53 


56 


10 


17 46 34 


1875 
lb/o 


2 19 


52 


53 


8 


4 


53 


34 


3 29 


42 


1 


10 


25 49 21 


20 11 


25 


33 


8 


23 


15 


46 


3 5 


19 


7 





4' 32 22 


1877 


10 2 


14 


10 


8 


12 


31 


38 


1 15 


7 


12 





12 35 9 


1878 


28 23 


46 


49 


9 





53 


50 


20 


44 


18 





21 18 10 


1879 


18 8 


35 


26 


8 


20 


9 


4211 


32 


23 


1 


29 20 57 


1880 
1881 


6 17 


24 


2 


8 


9 


25 


34| 9 10 


20 


28 


2 


7 23 44 


25 14 


56 


42 


8 


27 


47 


46j 8 15 


57 


34 


3 


16 6 45 


1882 


14 23 


45 


19 


8 


17 


3 


38 6 25 


45 


39 


3 


24 9 32 


1883 


4 8 


33 


55 


8 


6 


19 


30 5 5 


33 


44 


4 


2 12 19 


1884 


22 6 


6 


35 


8 


24 


41 


42 


4 11 


10 


50 


5 


10 55 20 


1885 


11 14 


55 


11 


8 


13 


57 


34 


2 20 


58 


55 


5 


18 58 7 


188b 


23 


43 


48 


8 


3 


13 


26 


1 


47 





5 


27 54 


1887 


18 21 


16 


28 


8 


21 


35 


38 


6 


24 


6 


7 


5 43 55 


1888 


7 6 


5 


4 


8 


10 


51 


3010 16 


12 


11 


7 


13 46 42 


1889 


26 3 


37 


44 


8 


29 


13 


42 9 21 


49 


17 


8 


22 29 43 


1890 
1891 


15 12 


26 


21 


8 


18 


29 


34J 8 1 


37 


22 


9 


32 30 


4 21 


14 


57 


8 


7 


45 


26' 6 11 


25 


27 


9 


8 35 IX 


1892 


22 18 


47 


37 


8 


26 


7 


38 4 17 


2 


33 


10 


17 18 ia 


1893 12 3 


36 


13 


8 


15 


23 


30 2 ' 26 


50 


38 


10 


25 21 5 


1894 


1 15 


24 


50 8 


4 


39 


22 1 6 


38 


43 


11 


3 23 52 


1895 


20 9 


57 


30 


8 


23 


1 


34! 12 


15 


49 





12 6 53 


1896 


8 18 


46 


6 


8 


12 


17 


2610 22 


3 


54 


O 


29 9 40 


1897 


27 15 


18 


47 


9 





39 


38 9 21 


41 





1 


28 52 41 


1898 


17 


7 


23 


8 


19 


35 


30 8 7 


29 


5 


2 


6 55 28 


' 1899 


6 8 


56 





8 


9 


11 


22 6 17 


17 


10 


2 


14 58 15 


1900 


24 6 


28 


40 8 


27 


33 


34| 4 22 


54 


16| 


3 


23 41 16 



AKITHMETICK. 



)?5 



TABLE III. Mean Anomalies^ and Sun 9 * mean Distance from 
the Nodcyfor 13-J mean Lunations. 


No. 

1 
2 
3 
4 
5 


Mean 
Lunations. 


Suns mean 
Anomaly. 


Moon's meari 
Anomaly 


bun's mean 

distance from 

the Node. 


D H. M. S 


s ' " 


s ' 


// 


s ' " 


29 12 44 3 

59 I 28 6 

88 14 12 9 

118 2 56 12 

147 15 40 15 


29 6 19 

1 28 12 39 

2 27 18 58 

3 26 25 17 

4 25 31 37 


25 49 

1 21 38 

2 17 27 

3 13 16 

4 9 5 




1 
1 
2 
2 


1 40 14 

2 1 20 28 

3 2 42 

4 2 40 56 

5 3 21 10 


6 
7 
8 
9 
10 


1*7 4 14 It* 
206 17 8 21 
236 5 52 24 
265 18 56 27 
295 7 20 30 


5 24 37 56 

6 23 44 15 

7 22 50 35 

8 21 56 54 

9 21 3 14 


5 4 54 

6 43 

6 26 32 

7 22 21 

8 18 10 


3 
3 
3 
4 
4 


6 4 1 24 

7 4 41 38 

8 5 21 52 

9 6 2 6 
10 6 42 20 


11 
12 
13 


324 20 4 33 
354 8 48 30 
38*3 21 32 40 


10 20 9 33 

11 19 15 52 
18 22 12 


9 13 59 

10 9 48 

11 5 37 


5 
5 
6 


11 7 22 34 

8 2 47 

1 8 43 1 




.14 18 22 2 


14 33 iO 


6 12 54 


3ul 


15 20 7 



156 



LOGARITHMICK 



1ABLE IV. The Day 9 of the Year, reckoned from the begin- 
ning of March. 


O 

to 


s 

p 

O 

ar 




2 

p 






> 
e 
<*? 
c 

r 


C/3 

B 

73 


O 

r* 
O 

cr 
71 


3 



< 

B 
0- 


7* 

246 
247 
248 
249 
250 


a 

rt 

B 



276 
277 
278 
279 
280 


*-• 
P> 
S 
C 
P 
~$ 

307 
308 
309 
310 
311 


cr 

e 

338 
359 
340 
341 
342 


1 

2 
3 
4 
5 


1 
2 
3 
4 
5 


32 

<■» r» 
OO 

34 
35 
36 


62 
63 
64 
65 
66 


93 
94 
95 
96 
97 


123 
124 
125 
126 
127 


154 
155 
156 
157 
158 


185 
186 
187 
188 
189 


215 
216 
217 
218 
219 


6 
7 

a 

9 
10 


6 
7 
8 
9 
10 


37 
38 
39 
40 
41 


67 
68 
69 
70 
71 


98 

99 

100 

101 

1C2 


128 
129 
130 
151 
132 


159 
160 
161 
162 
163 


190 
191 
192 
193 
194 


220 
221 
222 
223 
224 


251 
252 
255 
254 
255 


281 
282 
283 
284 
285 


312 
315 
314 
315 
316 


343 
344 
345 
346 
347 

448 
349 
350 
351 
352 


11 
12 
13 
14 
15 


11 
12 
13 
14 
15 


. 42 
43 
44 
45 
46 


72 
73 
74 
75 
76 


103 
104 
105 
106 
107 


133 
134 
135 
136 
137 


164 
165 
166 
167 
168 


195 
196 
197 
198 
199 


225 
226 
227 
228 
229 


256 
257 
258 
259 
260 


286 
287 
288 
289 
290 


317 
318 
319 
320 
321 


16 
17 
18 
19 
20 


16 
17 
18 
19 
20 


47 
48 
49 
50 
51 


77 
78 
79 
80 
81 


108 
109 
110 
HI 
112 


138 
139 
140 
141 
142 


169 
170 
171 
172 
173 


200 
201 
202 
203 
204 


230 
231 
232 
233 
234 


261 
262 
263 
264 
265 


291 
292 
293 
294 
295 

296 
297 
298 
299 
300 

301 
302 
503 
304 
305 
306 


322 
323 
324 
325 
326 

327 
328 
329 
330 
331 

332 
383 
334 
335 
336 
337 


353 
354 
355 
356 
357 

358 
359 
360 
361 
362 

363 
364 
365 
366 


21 
22 
23 
24 
25 

26 
27 
28 
29 
30 
31 


21 
22 
23 
24 
25 


52 
53 
54 
55 
56 


82 
83 
84 
85 
86 


113 
114 
115 

lie 

117 


143 
144 
145 
146 
147 


174 
175 
176 
177 
178 


205 
206 
207 
208 
209 


235 
236 
237 
238 
239 


266 
267 
268 
269 
270 


26 

27 
28 
29 
30 
31 


57 
58 
59 
60 
61 


87 
88 
89 
90 
91 
92 


lib 
W9 
120 
121 
122 


148 
149 
150 
151 
152 
153 


179 
180 
18i 
182 
185 
184 


210 
211 
212 
213 
214 


240 
241 
242 
243 
244 
245 


271 

272 
273 
274 
275 



ARITHMETICS. 



1ST 



TABLE V. Mean Lunations from 1 


to 100000. 




Lunat. 


Days Deci. parts. 


Days Hou. 


M. S. Th. 


Fo. 


1 


29-530590851080 


= 29 


12 


44 3 2 


58 


2 


59-061181702160 


59 


1 


28 6 5 


57 


3 


88-591772553240 


88 


14 


12 9 8 


55 


4 


118-122363404320 


118 


2 


56 12 11 


55 


5 


147-652954255401 


147 


15 


40 15 14 


52- 


6 


177-183545106481 


177 


4 


24 18 17 


50 


7 


206-714135957561 


206 


17 


8 21 20 


48 


8 


236-244726808641 


236 


5 


52 24 23 


47 


9 


265-775317659722 


265 


18 


36 27 26 


45 


10 


295-30590851080 


295 


7 


20 30 29 


43 


20 


590-61181702160 


590 


14 


41 59 


26 


30 


885-91772553240 


885 


22 


1 31 29 


10 


40 


1181-22363404320 


1181 


5 


22 1 58 


53 


50 


1476-52954255401 


1476 


12 


42 32 28 


36 


60 


1771-83545106481 


1771 


20 


3 2 58 


19 


70 


2067-14135957561 


2067 


3 


23 33 28 


2 


80 


2362-44726808641 


5362 


10 


44 3 57 


46 


90 


2657 75317659722 


2657 


18 


4 34 27 


29 


100 


29530590851080 


2953 


1 


25 4 57 


12 


200 


5906.1181702160 


5906 


2 


50 9 54 


24 


300 


, 88591772553240 


8859 


4 


15 14 51 


36 


400 


11812-2363404320 


11812 


5 


40 19 48 


48 


500 


14765-2954255401 


14765 


7 


5 24 46 





600 


17718-3545106481 


17718 


8 


30 29 43 


12 


700 


20671-4135957561 


20671 


9 


55 34 40 


24 


800 


23624-4726808^641 


23624 


11 


20 39 37 


36 


x 900 
1000 


26577-5317659722 


26577 


12 


45 44 54 


46 


29530-590851080 


29530 


14 


10 49 32 





2000 


59061-181702160 


59061 


4 


21 39 4 





3000 


88591-772553140 


88591 


18 


32 28 36 





4000 


1 18122*36*404320 


118122 


3 


45 18 8 





5000 


147652-954255401 


147652 


22 


54 7 40 


O 


6000 


177183-545106481 


177183 


13 


4 57 12 





7000 


206714-135957561 


206714 


3 


15 46 44 





8000 


236244-726801641 


236244 


17 


26 36 16 





9000 


265775-317659722 


265775 


7 


37 25 48 





10000 


295305-90851080 


295305 


21 


48 15 20 





20000 


59061 181702160 


590611 


19 


56 30 40 





30000 


885917 72553240 


885917 


17 


24 46 





40000 


1 188223-6S404320 


1188223 


15 


13 1 20 





50000 


1476529-542554U1 


1476529 


13 


1 16 40 





60000 


1771835-45106481 


1771835 


10 


49 32 





70000 


2067141-35957561 


2067141 


8 


37 47 20 





80000 


2362447-26808641 


2362447 


6 


25 2 40 





90000 


2657753-17659722 


2657753 


4 


14 18 


6 


100000 


2953959-0851080 


2953959 


2 


2 33 20 


o 



21 



i 



J58 



L06ARITHMICK 



~ _ . »-.- , i , i Ml. 1 1 

TABLE Y r I. The Jirst mean Aew Moott> with the mean Ano- 
malies of the Sun arrd Moon, and the Sun's mean Distance, , 
from the Ascending Nodc^ nezt after complete Centuries, of ' 
Julian Years. 



Luna- 
tions. 




First 1! 
Vcw Moon. 


Sun's meuti| 
Anomaly. 

s o '' 


Ylon'biiicaii 
Anomaly. 


Sun from 

Node. 


D.H. M.S.- 


so ' 


s 


O ' 


. 1237 1 
2474 
37U 
4948 
6185 
7422 
8658 
9895 


100 
200 
300 

400 


4' 8 10 52 

8 16 21 44 

13 32 37 

17 8 43 29 


3 2i 
« 42 
t) 10 3 
13. 24 


8 15 22 

5 44 

1 16 6 

10 1 28 


4 
9 
1 
6 


19 27 

8 55 

28 2% 

17 49 


500 
600 
700 
800 


21 16 54 21 

26 1 5 14 

20 32 3 

5 4 42 $5 


16 46 

20 7 

11 24 22 

11 27 34 


6 16 50 
3 2 12 

10 21 45 

7 7 7 


11 
3 
7 



7 16 
26 44 
15 31 

4 58 


11132 
12369 
15606 
14843^ 

16080 
17316 
18553 
19790 

21027 
22264 
23501 
24738 
25974 
27211 
28448 
29685 

3U922 
32159 
33396 
34632 


yoo 

1000 
1100 
1200 


9 12 53 47 
13 21 4 40 
18 5 15 32 
22 13 26 24 


14 
4 25 
7 46 
11 7 


3 22 29 
7 51 
8 23 13 
5 8 35 


4 
9 
2 
6 


V4 25 
13 53 

3 20 
22. 47 
12 15 

X 2 
20 29 

9 56 


1300 
1400 
1500 
1606 


26 21 37 16 
1 17 4 6 
6 1 14 58 

10 9 25 50 


14 28 
11 18 43 
11 22 4 
11 25 25 


1 23 57 
9 13 30 
5 28 52 

2 14 14 


11 
3 
7 



1700 
1800 
1900 
2000 


14 17 36 42 
19 1 47 35 
21 9 58 27 
27 18 9 19 


11 2<i 46 
2 8 
5 29 
8 50 


10 29 36 
7 14 58 
4 20 
15 42 


4 
9 
2 
6 


29 23 

18 51 

8 18 

27 45 


2100 
2200 
2300 
2400 


2 13 36 ^ 

6 21 47 1 

11 5 57 53 

15 14 8 45. 


11 13 5 
11 16 26 
11 19 47 
11 23 8 


8 5 15 
4 20 57 
1 5 59 

9 21 21 


10 
3 
7 



16 32 

6 

25 27 

14 54 


2500 
2600 
2700 
2800 


19 22 19 3H 

24 6 30 30 

28 14 41 22 

£ 10 8 11 


11 26 29 

11 29 50 

3 11 

11 7 76 


6 *6 43 

2 22 4 

11 7 26 

6 26 59 


■ 5 
9 
2 
6 


4 22 
23 4© 
13 16 

2 3 


35869 
37106 
38343 
39580 


2900 
3000 
3100 
3200 


7 18 19 3 

12 2 29 56 

16 10 40 48 

J 20 18 51 40 


11 10 47 
11 14 8 
11 17 30 
11 20 51 


3 12 21 
11 27 43 

- 8 13 5 

4 28 27 


10 

3 

8 

I 


21 30 
10 58 
25 
19 52 J 



ARITHMETICS 



15? 



TABLE VI. concluded. 




■ ■ !» 


V<J t-| 


Fust 


Sun 


*s mean 


Mon'smeanl Sun from 


Luna- 
tion a. 


S&.' 


New Moon- 


Anomaly. < 


. Anomaly. J Node. 


* g 


D. H. M. S 


s 


o 


t 


s 


o 


/ 


s 


O ' 


4081 7 


3300 


25 3 2 33 


11 


24 


12 


1 


13 


49 


5 


9 20 


42054 


3400 


29 11 13 25 


11 


27 


33 


9 


29 


11 


9 


28 47 


43290 


S50Q 


4 6 40 14 


11 


1 


48 


5 


18 


44 


1 


17 34 


44527 


3600 


8 14 51 6 


11 


5 


9 


2 


4 


6 


6 

10 


7 1 
26 29 


45764 | 


3700 


12 23 1 59 


U 


8 


30 


10 


19 


28 


47001 


3800 


17 7 12 51 


U 


11 


51 


7 


4 


50 


3 


15 56 


48238 


3900 


21 15 23 43 


11 


15 


12 


3 


20 


12 


8 


5 23 


49475 


4000 


25 23 34 35 


11 


18 


33 





5 


34 





24 50 


50/11 


4100 


19 1 27 


10 


22 


4o 


7 


25 


7 


4 


13 %r 


51948 


4200 


5 3 12 17 


10 


26 


9 


4 


10 


29 


9 


3 5 


53185 


4300 


6 11 23 *9 


-10 


29 


31 





25 


51 


1 


22 32 


54422 4400 


13 19 34 1 


11 


2 


52 


9 


11 


33 


6 


11 59 


55659 1 4500 


18 3 44 54 


11 


6 


13 


5 


26 


35 


11 


1 27 


56896 4600 


22 11 55 46 


11 


9 


34 


2 


11 


57 


3 


20 54 


58133 4700 


26 20 6 38 


11 


n 


55 


10 


27 


19 


8 


10 ?1 


59369 1 4800 


1 15 33 27 


10 
10 


17 


9 


6 


16 


52 


U 


29 8 


6U606 


4900 


5 23 44 20 


20 


21 


o 


2 


14 


4 


18 36 


61843 


5000 


10 7 55 12 


10 


23 


52 


li 


17 


30 


9 


8 3 


63080 


5100 


14 16 6 4 


10 


27 


13 


8 


2 


58 


1 


27 30 


64317 


5200 


19 16 56 


11 





34 


4 


18 


20 


6 


19 57 


65354 


5300 


23 8 27 49 


11 


3 


J>5 


1 


o 


42 


11 


6 25 


66791 


5400 


27 16 38 41 


id 


7 


16 


9 


19 


4 


2 


25 52 


68028 


5500 


2 12 5 30 


10 


11 


31 


5 


8 


S7 


7 


.14 39 


69265 


5600 


6 20 16 22 


10 


14 


52 


1 1 


23 


59 





4 6 

23 34 


70502 1 5700 


11 4 27 15 


10 


18 


14 


10 


9 


21 


71739 5803" 


15 12 38 7 10 


21 


35 


6 


24 


43 


9 


13 1 


72976 5900 


19 20 48 59 10 


24 


56 


3 


10 


5 


2 


2 28 


74212 6000 24 4 59 52 1 10 


28 


17 U 


25 


27 


6 


21 56 


' h Ov. found mean Lunation (which we 


* have kept 


uy in 


:nakin£ these tables) be added 74212 times 


to itself, the 


sum will amount to 6ooo Julian years, 24 


days 4 hours 59 


mnutes 51 seconds 4o thirds 


; as 


reeing with the first part 


>Fthe iast line of this table, with 


n halt' a 


second. 




■ "r't 



t6* 



LOGARITHMICK 



TABLE Vll. The annual) or Jirttt Equation of the mean to the 
true Syzygy. 




Argument 


. Sun's mean Anomaly. 






Subtract. 


a 

(ft 

CD 



Signs. 


1 
Sign. 


1 2 
Signs. 

H. M.S 


3 

Signs. 


4 
Signs. 


5 
Signs. 


S? 

n 

CO 

15 

29 
28 
27 
26 
25 

T4 
23 
22 
21 
20 


H. 


M.S 


H.M S 


H.M.S 


H. MS 


H. M. S. 











2 3 12 


3 


35 


4 10 53 


3 39 30 


2 7 45 


1 

2 
3 
4 
5 









4 18 

B35 

12 51 

17 8 

21 24 


2 6 do 
2 10 36 
2 14 14 
2 17 52 
2 21 27 


3 

3 
3 
3 
3 


37 10 
39 18 
41 23 
43 26 
45 25 


4 10 57 
1 4 10 54 
4 10 49 
4 10 39 
4 10 24 


3.37 19 
3 35 6 
3 32 50 
3 30 30 
3 28 5 


i 3 55 
2 1 
1 56 5 
1 52 6 
1 48 4 


6 

8 

9 

10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 









25 39 
28 55 
34 11 
36 26 
42 39 


2 25 9 
2 28 29 
2 31 57 
2 35 22 
2 38 44 


3 
3 
3 
3 
3 


47 19 

49 7 

50 50 
52 29 
54 4 


4 10 4 
4 9 39 
4 9 10 
4 8 37 

4 7 59 


3 25 35 
3 23 
3 20 20 
3 17 35 
3 14 49 


1 41 1 
1 39 56 
1 35 49 
1 31 41 
1 27 31 







1 


46 52 
51 4 
55 17 
59 27 
3 36 


2 42 3 
2 45 18 
2 48 30 
2 51 40 
2 54 48 


3 

3 
3 
3 
3 


55 35 

57 2 

58 27 

59 49 
1 7 


4 7 16 
4 6 29 
4 5 37 
4 4 41 
4 3 40 


3 11 59 
3 9 6 
3 6 10 
3 3 10 
3 7 


1 23 19 
1 19 5 
1 14 49 
1 10 33 
1 6 15 

1 1 56 
57 36 
53 15 
48 52 
44 28 


19 
18 
17 
15 
16 
14 
13 
12 
11 
10 

9 
8 
7 
6 
5 
4 
3 
2 
1 


O 
« 


1 
1 
1 

1 
1 


7 45 
11 53 
16 
20 6 
24 10 


2 57 53 

3 54 
3 3 51 
3 6 45 
3 9 36 


4 
4 
4 
4 
4 


2 18 

3 23 

4 22 

5 18 

6 10 


4 2 35 
4 1 26 
4 12 
3 58 52 
3 57 27 


2 57 
2 53 49 
2 50 36 
2 47 18 
2 43 57 


21 
22 
23 

* 24 
25 


1 
1 
1 

1 

1 


28 12 
32 12 
36 10 
40 6 
44 1 


3 12 24 
3 15 9 
3 17 51 
3 20 30 
3 23 5 


T 

4 
4 
4 
4 


6 58 

7 41 

8 21 

8 57 

9 29 


3 55 59 
3 54 26 
3 52 49 
3 51 9 
3 49 26 


2 40 33 
2 37 6j 
2 33 35 
2 30 2 
2 26 26 


40 2 
35 36 
31 10 
26 44 
22 17 


26 
27 
28 
29 

\ 30 

°i 
a 
00 


1 
1 
1 
1 

2 


47 54 
51 46 
65 37 
59 26 
3 12 


3 25 36 
3 28 3 
3 30 26 
3 32 45 
3 35 


4 

4 
4 
4 

4 


9 o5 
10 16 
10 33 
10 45 
10 53 


3 47 38 
3 45 44 
3 43 45 
3 41 40 
3 39 30 


2 22 47 
2 19 5 
2 15 20 
2 11 35 
2 7 45 


17 50 
13 23 
8 56 
4 29 



11 

Signs. 


10 
Signs. 


9 
Signs. 


8 
Signs. 


7 
Signs. 


6 
Signs. 


Add |, 



ARITHMETICS 



161 



iAtiJLk Vlli. 


Aquation of the 


Muon y 8 mean Anomaly. 


Argument. Sun's mean Anomaly. 


Subtract. 


O 





1 




2 


3 




4 




i> 


O 


OB 


Signs. 


Signs. 


Signs. 


Sign 


s. 
n 


Signs. 


Signs. 


o» 


o ' '' 


' 


n 


' " 


' 


' 


// 


' '' 








46 


45 


1 21 32 


1 35 


ll 1 23 


4 


48 19 


30 


1 


1 37 


48 


10 


1 22 21 


1 35 


2 1 22 


14 


46 51 


29 


2 


3 13 


49 


34 


1 23 10 


1 35 


1 


1 21 


24 


45 23 


28 


3 


4 52 


50 


53 


1 23 57 


1 35 





1 20 


32 


43 54 


27 


4 


6 28 


52 


19 


1 24 41 


1 34 


57 


1 19 


38 


42 24 


26 


5 
6 


08 6 


53 


40 1 25 24 


1 34 


50 


1 18 


42 


40 53 


25 


9 42 


55 





1 26 6 


1 34 


43 


1 17 


45 


39 21 


24 


r 


Q 11 20 


56 


21 


1 26 -48 


1 34 


33 


1 16 


48 


37 49 


23 


8 


12 56 


57 


38 


1 27 28 


1 34 


22 


1 15 


47 


36 15 


22 


9 


14 33 


58 


56 


1 28 6 


1 34 


9 


1 14 


44 


34 40 


21 


10 
11 


16 10 


1 


13 


1 28 43 


1 33 


53 


1 13 


41 


33 5 


20 


17 47 


1 1 


29 


1 29 17 


1 33 


37 


1 12 


37 


31 31 


19 


1? 


19- 23 


1 2 


43 


1 29 51 


1 33 


20 


1 11 


33 


29 54 


18 


13 


20 59 


1 3 


56 


1 30 22 


1 33 





1 10 


26 


28 18 


17 


14 


22 35 


1 5 


8 


1 30 50 


1 32 


38 


1 9 


17 


26 40 


16 


15 
16 


24 10 


1 6 


18 


131 19 


1 32 


14 


1 8 


8 


25 3 


1* 

14 


25 45 


1 7 


27 


1 31 45 


1 31 


50 


1 6 


58 


23 23 


17 


27 19 


1 8 


36 


1 32 12 


1 31 


23 


1 5 


46 


21 45 


13 


18 


28 52 


1 9 


42 


1 32 34 


1 30 


55 


1 4 


32 


020 7 


12 


19 


30 25 


1 10 


49 


1 32 57 


1 30 


25 


1 3 


19 


18 28 


11 


20 
21 


31 57 


1 11 


54 1 33 17 


1 29 


54 


1 2 


1 


16 48 


10 
9 


33 29 


1 12 


58 1 33 36 


1 29 


20 


1 


45 


15 8 


22 


35 2 


1 14 


1 


133 52 


1 28 


45 


59 


26 


13 28 


8 


23 


36 32 


1 15 


1 


1 34 6 


1 28 


9 


58 


7 


11 48 


7 


24 


38 1 


1 16 





1 34 18 


1 27 


30 


56 


45 


10 7 


6 


25 

26 


39 29 


1 16 


59 


1 34 30 


1 26 


50 


55 


23 


8 20 


5 

4 


40 59 


1 17 


57 


1 34 40 


1 26 


27 


54 


r 


6 44 


27 


42 26 1 18 


52 


1 34 48 


1 25 


5 


52 


37 


5 3 


3 


< 28 


43 54 


1 19 


47 


1 34 54 


1 24 


3$ 


5i 


12 


3 21 


2 


29 


45 19 


1 20 


40 


1 34 58 


2 23 


52 


49 


45 


1 40 


1 


30 

O 


47 45 


1 21 


32 


1 1 35 1 


1 23 


4 


48 


19 





_0 

a 


i 1 


1 




9 


8 




7 




6 


to 
TO 


St^ns. 


Sinrns. 


Sifrns. * Sien*. Sign 


s. Signs. 




AUU 



Sfrl 



LOGARITHMICK 



i AiiLli IX.. iVitf aecond Jiquuuuii of' urn mean la me trht" 

Syzygy. 


Argument. Moon's Equated Anornaiy. 


Add 


O 

o 

ft 

a 

en 



Signs. 


Sign. 


Signs. 


3 
Signs. 


4 
Signs. 


5 

Signs. 


O 
eg 
ci 


H. M. S. 


rf. M. S. 


H. M. S. 


H.M.S. 


H.M. 


S 


H. M. S. 


J» 


o u 


5 12 4tf! 8 47 8 


9 46 44| 8 8 


59 


4 34 33 


30 


J 

5 

6 

7 

" 8 

9 

10 

11 
12 
13 
14 
15 

♦ 16 
17 
18 
19 

20 

21 
22 
23 
24 
25 

26 
27 
28 
29 
30 

C 
c 


10 d6 
21 56 
32 54 
42 52 
54 50 


5 21 50 
5 30 57 
5 39 51 

5 48 37 
5 57 17 


8 51 45 

8 56 10 

9 25 
9 4 31 
9 8 25 


9 45 3 
9 45 12 
9 44 11 
9 42 59 
9 41 36 


8 3 
7 57 
7 51 
7 45 
7 39 


12] 
23 
33 
46 
46 


4 26 1 
4 17 25 
4 8 47 
4 7 
3 .1 23 


29 
28 

27 

26 
25 

24 
23 
22 
21 
20 

19 

18 
17 
16 
IS 

14 

ia 

12 

li 

10 

9 
8T 
7 
6 
5 

4 
3 
2 
1 



1 5 48| 6 5 oi 
1 16 46 6 14 19 
1 27 44 6 22 41 
1 38 401 6 30 57 
1 49 33| 6 39 4 


9 12 9 
9 15 43 
9 19 5 
9 22 14 
9 25 12 


9 40 3 
9 38 19 
9 36 24 
9 34 18 
9 32 1 


7 33 
7 27 
7 21 
7 14 
7 7 


36 

22 
o 

30 
50 


3 42 32 
3 33 38 

2 24 42 

3 15 44 
3 6, 45 


2 2o 
2 11 10 
2 21 54 
2 32 34 
2 43 9 


6 47 

6 54 46 

7 2 24 
7 9 52 
7 17 9 


9 2/ 58 
9 30 32 
9 32 58 
9 35 12 
9 37 14 


9 29 33 
9 26 54 
9 24 4 
9 21 3 
9 17 51 


7 1 
6 54 
6 47 
6 40 
6 32 


2 
8 
9 
6 
56 


2 57 43 
2 48 39 
2 39 34 
2 30 28 
2 21 19 


2 53 3d 

3 4 3 
3 14 24 
3 24 42 
3 34 58 


7 24 10 
7 31 18 
7 38 9 
7 44 51 
7 51 24 


9 39 8 
9 40 51 
•9 42 21 
9 43 42 
9 44 53 


9 14 28 
9 10 54 
9 7 9 
9 3 13 
8 58 6 


6 25 
6 18 
6 10 
6 3 
5 55 


40 
18 
49 
16 
38 


2 tit 8 
2 2 53 
1 53 36 
1 44 16 
1 34 54 


3 45 11 

3 55 21 

4 5 26 

4 25 26 
4 25 20 


7 57 45 

8 3 56 
8 9 57 
8 15 46 
8 21 24 


9 45 '52 
9 46 38 
9 47 13 
9 47 36 
9 47 49 


8 54 50 
8 50 24 
8 45 48 
8 41 2 
8 36 6 


5 47 
5 40 
5 32 
5 24 
5 16 


54 
4 
9 
9 
5 


1 25 31 
1 16 7 
1 6 41 
57 13 

47 44 


4 35 6 
4 44 42 

4 54 11 

5 3 33 
5 12 48 


8 €6 53 
8 32 11 
8 37 19 
8 42 18 
8 47 8 


9 47 54 
9 47 46 
9 47 32 
9 47 14 
9 46 44 

9 
Signs. 


8 31 
8 25 44 
8 20 18 
8 14 33 
8 8 59 


5 7 

,4 59 
4 51 
4 43 
4 34 


56 
42 
15 

2 
33 

s. 


38 13 
28 41 
19 8 
9 34 



•1 I 
S ; gns. 


10 
Signs. 


8 
Signs. 


7 
Sign 


6 J "a 

Signs. 1 « 








Sub 


tract 











ARITHMETICK. 



169: 





LIABLE X. the third Equa- 


TABLE XL The fourth Aqua- 




, 


tion of the mean to the true 


tion of the mean to the true 






Syzygy. 


Syzygy; t 




Argument. Sun's Anomaly. 


Argument. Sun's mean distance 






Moon's Anomaly 


from the Node. 






o 

a> 
n 
••* 


Signs. |Signs. (Signs. 


a 

79 


/■via 






Sub. 
6 Add 


1 Sub. 
7 Add 


2 Sub. 
8 Add 


"o 


}{sig.|!{.Si B |^S«K.|0 

.. '. - ' -- - ; i era 






CO 

6 


M. S. |M- S. 


M. S. 


To 


i\l. S. | iYl. ^. 1 ivl. 5. 


30 









2 22 


4 12 





1 22 


1 22 






1 


5 


2 26 


4 16 


29 


1 


4 


1 23 


1 21 


29 






2 


10 


2 30 


4 18 


28 


2 


7 


1 24 


1 20 


28 






3 


15 


2 34 


4 21 


27 


3 


10 


1 25 


1 18 


27 






4 


20 


2 38 


4 24 


2b 


4 


13 


1 26 


1 16 


26 






5| 


25 


2 42 


4 27 


25 


5 
6 


16 


1 27 


1 14 


25 
24 






61 30 


T 46 


4 30 


24 


20 


1 28 


1 12 






7 35 


2 50 


4 32 


23 


7 


23 


1 29 


1 10 


23 






8 


40 


2 54 


4 34 


22 


8 


26 


1 SO 


1 8 


22 






9 


45 


2 58 


4 36 


21 


9 


29 


1 31 


1 6 


21 






10 
11 


50 


3 2 


4 38 


20 


10 
11 


32 


1 32 1 3 


20 
19 






65 


3 6 1 


t 4 40 


19 


35 


1 33 | 1 






12 


1 


3 10 


4 42 


18 


12 


38 


1 33 57 


18 






13 


1 5 


3 14 


4 44 


17 


13 


41 


1 34 


54 


17 






14 


1 10 


3 18 


4 46 


16 


14 


44 


1 34 


51 


16 






15 
16 


1 15 | 3 22 


4 48 


15 


15 


47 


1 34 


49 1 15 






1 20 


3 26 


4 50 


14 


16 


50 


1 34 


45 


14 






17 


1 25 


3 30 


4 51 


13 


17 


52 


1 34 


41 


13 






18 


1 30 


3 34 


4 52 


12 


IB 


54 


1 34 


37 


12 






19 


1 35 


3 38 


4 53 


11 


19 


57 1 33 


34 


11 




1 20 
^21 


|'l 40 


3 42 


4 54 


10 


20 
"21 


1 1 1 33 


31 


10 




i 1 45 1 3 45 


4 55 


9 


1 2 


1 32 


28 


9 






22 1 49 1 3 48 


4 56 


8 


22 


1 5 


1 31 


25 


8 






23 1 52 3 51 


4 57 


7 


23 


1 8 


1 30 


22 


7 






24 1 56 3 54 


4 57 


6 


24 


1 10 


1 29 


19 


6 






25 2 j 3 57 4 57 


5 


25 


1 12 


1 28 


16 


5 






26! 2 4 i 4 


4 58 


A 


F26 


1 14 


1 27 


13 


4 






27 2 9 4 3 


4 58 


t 


27 


r 1 16 


1 26 


10 


3 






28 2 13 4 6 


4 58 


c 


> 2S 


I 1 18 


1 25 


6 


2 






29 2 18 1 4 9 


4 58 


\ 


^ 2< 


) 1 20 


1 24 


3 


1 






O 
'2 

10 


) 2 22 1 4 12 


4 58 


( 


)| 3( 


) 1 22 


1 22 












Signs. [Signs. (Signs 


? 


:0 

O' 
CO 


>^ si 


M* 


M** 


o 

en- 


I 




5 Suu.l 4 Sue 


. 3 Sub 

d9 Ad 








11 Ad 


djlOAd 


Subtrac 


t 


1 2 

CO 


't 



i 



144 



LOGARITHMICK 



a TABLE XII. The Sun's mean Longitude* Motion* and 


1 


Anomaly : 9ld Style. 


ft 


Years 
beginning 


Sun's mean 
Longitude. 


Sun's mean' 
Anomaly, j 


o 

2 < 
■2.5 

r* 

CD 


Sun's mean 
Motion. 


Suns mean 
Anomaly. 


s o ' " 


s o ' 1 


s 


o ' " 


so ' 


l 


9 7 53 10 


6 28 48 


19 


11 


29 24 16 


11 29 4 


201 


9 9 23 50 


6 26 57 


20 





9 4 


11 29 48 


301 


9 10 9 10 


6 26 I 


40 





18 8 


11 29 37 


401 


9 10 54 30 


6 25 5 


60 

> 





27 12 


11 29 26 


501 


9 11 39 50 


6 24 9 


80 





36 16 


1 1 29 15 


1001 


9 15 26 SO 


6 19 32 


100 





45 20 


11 29 4 


1101 


9 16 11 50 


6 18 36 


200 





1 20 40 


11 29 8 


1201 


9 16 57 10 


6 17 40 


300 





2 16 


il 27 12 


1301 


9 17 42 30 


6 16 44 


400 


9 


3 1 20 


11 26 16 \ 


1401 


9 18 27 50 


6 15 49 


500 





3 46 40 


11 25 21 


1501 


9 19 13 10 


6 14 53 


600 





4 32 


11 24 25 


1601 


9 19 58 30 


6 13 57 


700 





5 17 20 


11 23 29 


1701 


9 20 43 50 


6 13 1 


800 





6 2 40 


11 22 33 


1801 


9 21 29 10 


6 12 6 


900 
1000 






6 48 

7 33 20 


11 21 37 
11 20 11 








•as 


Sun's Mean 
Motion. 


Sun's mean 
Anomaly. 


2000 
3000 
4000 
5000 
6000 





1 
1 

1 


15 6 40 

22 40 

13 20 

7 46 40 

15 20 


11 11 22 
11 2 3 
10 22 44 
10 13 25 
10 4 6 


s o ' " 


so' 


1 

2 

3 

4 


11 29 45 40 

11 29 31 20 

11 29 17 

1 49 


1 1 29 45 
11 29 29 
11 29 14 
11 29 58 
11 29 42 


2 

o 

a 

*-* 
cr 

wo 


Sun*s mean 
Motion. 


Suns mean 
Anomaly. 


5 


11 29 47 29 








6 

7 


11 29 33 9 
11 29 18 49 


11 29 27 
11 29 11 




s 


o ' " 


so' 


Jan. 





O 





8 


3 38 


11 29 5 


Feb. 


i 


33 18 


1 33 


9 


11 29 49 18 


11 29 40 


Mar. 


l 


28 9 11 


1 28 9 


io 


11 29 34 58 


11 29 24 


Apr. 


2 


28 42 30 


2 28 42 


11 


11 29 20 38 


11 29 9 


May 


3 


28 16 40 


3 28 17 


12 


5 26 


11 29 53 


June 


4 


28 49 58 
28 24/ 8 


4 28 50 


13 


11 29 51 7 


rl 29 37 


July 


5 


5 28 24 


14 


11 29 36 47 


11 29 22 


Aug 


6 


29 57 26 


6 28 57 


15 


11 29 22 27 


11 29 7 


Sepi 


7 


29 30 44 


7 29 30 


16 


7 15 


11 29 50 


Oct. 


8 


29 4 54 


8 29 4 


17 


11 29 52 55 


11 29 35 


Nov 


9 


29 38 12 


9 29 37 


■jr is 


12 29 38 35 


11 29 20 


Dec 


10 


29 12 *>2 


10 29 11 



AitlTHMfcTICK. 



165 







TABLE 


XII. 


concluded. 






; 


j S«i 


n's 


mean i Sun's 


mean's. 


mean Sun's 


mean 


S. mean 


Motion and 


Motion and 


Dist. fro. Motion and 


distance 




Anomaly. 


Anomaly. 


Node 




Anomaly. 


fro.Node. 


H 10 

m|/ 




tt 


' 


H « 


/ 


1 




II 




s 


o 


i It 


a 


ttl. 1 

If It ■- 


fi 


'"M 


r 


11 


tn\ ' // 


III 


1 

2 






U^ 59 

1 58 17 


S " 
"~io 


2 


2i 


rr 

\ 


iff 


ini 

•■ 1 


5 

Ti 




in 


It ft ". H> 


It', 





2 


36 


1 


16 


231 20 


30 


3 





2 


57 25 


20 


4 


560 


5 


12i 30 


1 


18 


5lll 23 


6 


4 





3 


56 33 


30 


5 


240 


7 


48j 33 


1 


21 


19,1 25 


42 


5 





4 


55 42 


40 


510 


10 


23 34 


1 


23 


471 28 
151 30 


18 


6 





5 


54 50 


50 


12 


190 


12 


50 


35 


1 


26 


54 


7 





6 


53 58 


60 


14 


470 


15 


35 


36 


1 


28 


421 33 


29 


8 





7 


53 7 


7!0 


17 


150 


18 


11 


37 


1 


31 


io'i 36 


5 


9 





8 


52 15 


80 


19 


430 


20 


47 


38 


1 


33 


38'1 38 


40 


10 





9 


51 23 


90 


22 


110 


23 


23 


39,1 


36 


61 41 


16 


11 





10 


50 32 


100 


24 


380 


25 


58 


40!1 


38 


34'1 43 


52 


12 





11 


49 40 


110 


27 


60 


28 


34 


411 


41 


21 46 


28 


13 





12 


48 48 


120 


29 


340 


31 


10 


42'1 


43 


30ll 49 


4 


14 





13 


47 57 


13.0 


32 


20 


33 


45 


43 : 1 


45 


57J1 51 


39 


15 





14 


47 5 


HO 


34 


360 


36 


21 


44 j l 


48 


251 54 


15 


16 





35 


46 13 


15*0 


36 


580 


38 


57 


45; 1 


50 


531 55 


51 


17 





16 


45 22 


16 ! 


39 


260 


41 


33 


46,1 


53 


211 59 


27 


18 





17 


44 30 


170 


41 


53;0 


44 


8 


47 1 


55 


49 2 2 


3 


19 





18 


43 38 


180 


44 


21Q 


46 


44 


481 


58 


172 4 


39 


20 





19 


42 47 


19.0 


46 


49,0 


49 


!20 


49,2 





442 7 


13 


1 21 





20 


41 55 


20:0 


49 


17>0 


51 


56 


502 


3 


12,2 9 


50 


22 





21 


41 3 


21|0 


51 


45*0 


54 


32 


512 


5 


40!2 12 


25 


23 





22 


40 12 


22|0 


54 


130 


57 


8 


522 


.8 


8'2 15 


2 


24 





23 


39 20 


23J0 


56 


40.0 


69 


43 


532 


io 


36 


2 17 


38 


25 





24 


38 28 


24,0 


59 


8 ! 1 


i 


19 


&2 


13 


4 


2 20 


14 


26 





25 


37 37 


25J1 


1 


361 


4 


55 


552 


15 


32 


2 22 


50 


27 





26 


36 45 


261 


4 


41 


7 


31 


56 2 


17 


59 


2 25 


26 


28 





27 


35 53 


271 


6 


321 


10 


7 


57i3 


20 


27 


2 28 


8 


29 





28 


35 2 


28;i 


9 


01 


12 


43 


582 


22 


55 


2 3o 


32 


30 


29 


34 10 


29<f 


11 


28'1 


15 


19 


592 


25 


23 


2 33 


14 


31' 1 


30 


33 18 


301 


13 


551 


17 


55 


602 


27 


51 


2 35 


50 


In leap years, alter February, add one day, au< 


1 oho clay's motion. 



22 



1 



166 



LOGARITHMICK 



i'AbLli XI 11. liquation of the Sun's Centre, or the Differ- 






ence between his mean and true Place, 






Argument. Sun's mean Anomaly. 


Subtract. 


^j 


O 


t i a 


3 


4 


5 O 


>3 £ 

r 

rt 


Signs 


9 


Sign. 


Signs. 

o * * 


Signs. 


Signs. 


Signs. 


o 

-s • 

n 

o 

C0 

30 
29 


o ' 


' " 


' " 





"1 





t 


56 47, 1 39 6 


1 55 37| 1 41 12 


58 53 


1| 


1 


59 


58 30 1 40 7 


1 55 39 


1 40 12| 57 7 


2; 3 


57 


1 12, 1 41 6 


1 55 38 


1 39 10 55 19 


28 


3 


5 


56 


1 1 53 1 42 3 


1 55 36 


1 33 6 53 30 


27 


4 


7 


54 


1 3 33 1 42 59 


1 55 31 


1 37 


51 40 


26 


5 

6 


9 


52 


1 5 12j 1 43 52 1 55 24 


1 35 52 


49 49 


25 
24 


u 


50 


1 6 50 1 44 44J 1 55 15 


1 34 43 


47 57 


7 


13 


48 


1 8 27i 1 45 34 


1 55 S 


1 33 32 


46 5 


23 


8 


15 


46 


1 10 2| 1 46 22 


r 54 50 


1 32 19 


44 11 


22 


9 


17 


43 


1 11 36| 1 47 8 


1 54 35 


1 31 4 


42 16 


21 


10 19 


40 


1 13 9| 1 47 53 


I 54 \7 


1 29 47 


40 21 


20 
19 


11 


21 


37 


1 14 411 1 48 36 


1 53 57 


1 23 29 


38 25 


12 


23 


oo 


1 16 11 1 49 15 


1 53 36 


1 27 9 


36 28 


18 


13 


25 


29 


1 17 40 1 49 54 


1 53 12 


1 25 48 


34 30 


17 


14 


27 


25 


1 19 8 1 50 30 


1 32 46 


1 24 25 


32 32 


16 


.15 
16 


29 
31 


20 
15 


1 20 34) 1 51 5 


1 52 18 


1 23 


30 33 


15 

14 


1 21 59 


1 51 37 


1 51 48 1 21 34 


28 33 


17 


33 


9 


1 23 22 


1 52 8 


1 51 15 


1 20 a 


26 33 


13 


18 


35 


2 


1 24 44 


1 52 36 


1 50 41 


1 18 36 


24 33 


12 


19 


36 


55 


1 26 5 


1 53 8 


1 50 5 


1 17 $ 


► 22 32 


11 


20 


38 


47 


1 27 24 


1 53 27 


1 49 26 1 15 3^ 


20 30 


10 
9 


21 


40 


39 


1 28 41 


1 53 5Q 


1 48 46 


1 13 5S 


18 28 


22 


42 


30 


1 29 57 


1 54 10 


1 48 3 


1 12 24 


• 16 26 


8 


23 


44 


20 


1 31 11 


1 54 28 


1 47 19 


1 10 47 


14 24 


7 


24 


46 


9 


1 32 25 


1 54 44 


1 46 32 


19 9 


12 21 


6 


23 
26 


47 


57 


1 33 35 


1 54 58 1 45 44 


1 7 29 


10 18 


5 

4 


> 49 


45 1 34 45 


i 55 10| 1 44 53| I 5 4S 


> 8 14 


( 27 


' 51 


3? 


1 35 53 


1 55- 20 


1 44 11 1 4 7 


6 11 


3 


28 


53 


18 


1 36 59 


1 55 28 


1 43 7 1 2 24 


4 7 


2 


2S 


55 


3 


1 38 3 


1 55 34 


1 42 10 1 1 39 


2 4 


1 


30 


56 


47 


1 39 6 


1 55 37 


1 41 12 


58 55 







O 


1 1 




iw 


9 


8 


7 


6 


09 


Si.rns. 


S»Jrr.s. 


Signs. 


Si >ns. 


Signs. 


Signs. 


•7Q 


Auu 



ARITHMETICK 



16/ 



r ABLE XIV. The 


Hun's 


TABLE XV. Equation oj 


'the &nnu 




Declination. 




\ie 


mean Distance from the 


Node. 


Argument. 


Sun's 


tr 


Argument, aun J s> meun Auomaiy. 




Place. 


















Sis 


rib 


Si^ns 


Signs 


- 






Su 


tbtract. 




Cf5 





vV. 


1 


j\\ 


2 


j\r. 


n 







I 


2 


5 




4 1 5 


O 




6 


s 


7 


S. 


8 

"~o" 

2(T 
20 


» 

Ti 

24 


6 
? 1 

30 

29 


CO 


Sip: 


Si* 


Sl£. 


Si 


'A* 


S 


>£• 


Si*. 








IT 


> 





. 3 


o ' 


O ' 


' 





i° 


' 


* 





11 
11 


30 
51 


"o'o 


01 
2;1 


2 ; 

4 


1 


4/2 


5!1 


50 


1 4 


30 


1 





24 


"10 


1 


48 2 


5!,1 


48 


1 2 


29 


2 





48 


12 


11 


20 


36 


23 


20 


41 


61 


49 


2 


51 


47 


1 


28 


3 


l 


12 


12 


32 


20 


48 


27 


3 


6!l 


81 


50 


2 


51 


46 


58 


27 


4 


l 


36 


12 


53 


20 


59 


26 


40 


9 


l 


101 


51 


2 


51 


45 


56 


26 


5 


l 


59 


13 
13 


13 
33 


21 


10 


25 

94 


5 
6 





11 


l 


121 


52 


2 


5;1 


44 


54 


25 
24 


6 


2 


23 


21 


21 





13 


l 


141 


53 


2 


51 


43 


52 


7 


2 


47 


13 


53 


21 


31 


23 


7 


1^1 


161 


54 


2 


4'1 


41 


50 


23 


8 


3 


11 


14 


12 


21 


41 


9? 


80 


171 


171 


55 


2 


4!l 


40 


48 


22 


9 


3 


3414 


31 


21 


50 


91 


p0 


191 


181 


56 


2 


4 \ l 


39 


46 


21 


10 3 


58 


14 


50 


21 


59 


20 
19 


10 o 
Tio 


211 
231 


19 
'21 


1 


57 


2 


4|7 


37 


44 


20 


11 


4 


22 


15 


9 


22 


8 


1 


85 


2 


3jl 


3b 


042 


19 


12 


4 


45 


15 


28 


22 


16 


18 


120 


251 


221 


58 


2 


31 


34 


40 


18 


13 


5 


9 


15 


1 


22 


24 


17 


13|0 


281 
3011 


241 


59 


2 


31 


33 


37 


17 


14 


5 


3216 


22 


3X 


16 


14° 


26|2 





2 


21 


31 


35 


16 


15 5 


35 


16 


22 


22 


38 


15 

14 


15 
! 16 





321 


27l2 





2 


2|1 


30 


33 


15 
14 


16 6 


18 


16 


39 


22 


45 





34 


l 


28j2 


1 


2 ' 


1]1 


28 


31 


17 


6 


41 


16 


57 


22 


51 


T3 


' 17 
| 18 j 


36 


1 


■30J2 


1 


2 


11 


27 


29 


13 


18 


7 


4 


17 


14 


22 


56 


12 


381 


31 


2 


2 


2 


01 


25 


27 


12 


19 


7 


27 


17 


30 


23 


2 


11 


19!0 


401 


34 


2 


2 


2 


01 


24 


24 


11 


20 


7 


50 


17 


46 


23 


6 
11 


10 


! 20 
! 21 


'0 


42J1 


35 


2 


3 


1 


59|1 


23 


22 


10 

9 


21 


8 


15 


18 


2 


23 


.0 


44 ! 1 


36 


2 


3 


1 


59:1 


21 


20 


22 


8 


35 


18 


18[23 


14 


8 


! 920 


46' 1 


37 


2 


41 


581 


19 


18 


8 


23 


9 


57 


18 


33i23 


18 


7 


I 23 


48| 1 


39 


2 


4jl 


57! 1 


17 


16 


7 


24 


9 


20 


18 


48*23 


21 


6 


1 24'0 


50|1 


40 


2 


41 


561 


15 


13 


6 


25 
26 


9 


42 


19 


3 ! ,23 


21 


5 

4 


1 25',0 

i ™P 


52 
54 


11 


41 


2 


4|l 


551 


13 


11 


5 

"4 


10 


4 


19 


1723 


25 


;1 


43J2 


51 


541 


11 


9 


27 


10 


25119 


3123 


27 


3 


27 
28!0 


561 


442 


5 


1 


531 


9 


7 


3 


28 


10. 


47|19 


4523 


28 


2 


58 ! 1 


45;2 


5 


1 


521 


8 


5 


2 


29 


11 


819 


58|23. 


29 


1 


2911 


o;i 


462 


5 


1 


511 


6 


3 


1 


30 

a 


11 


30 20 


Ilj23 


29 





30 1 


2 ! 1 


47!2 


5 


1 


50 1 


4 


0! Q 


Signs) 


Si 


tf.i& 


Si| 


-lib 


a 




l 1 




10 1 


9 




8 1 


7 


6 



















-5 

o 


re 


Q 




*j 


l£. J S 




1 S 


iff-' *" 


ix r 


M »• 




\\ 


5 


10 


5 


9 


&' 








3 i 












5 5 


jv. 4 j\r. 3 


j\r 


<*5 


» 






Add 






, ^ 



16$ 



LOGARITHMICK 



TAliLK XVI. 


TAbLE XV11. The iuuon'tt horizontal Parallax 


The Aloon's 


with the Semidiametera 


and true Horary Motion 


Latitude in 


of the Sun and Moon y and every sixth Degree 


q/ 


£ctifl8f8. 


their mean Anomalies y the Quuntitits for the in* 
ter mediate Degrees being easily proportioned by 


Argument. 


-Moon's equa- 
ted Distance 


sight. 


















' 


















from Node. 


S8> 

g 09 | 

=> c 3 

* a £. 

2 - 


T3 3" 
sT Q 

*L N 
• P 


c 
• o 

UB* 


S3 I 

JS* (A* 

i* 


SI* 
3 


2K2 

s s s 

2 ~» ^ 


s s 

O -1 




P § I 

D 


Sighs. 
Aforth ascend. 


South UeoCtna 





V 


u 


s o 


/ 


/' 


-/ " 


/ 


// 


f ~ 


ft 


'■>'* 


" js 


o 



1 


Q 
5 15 


30 
9,9 


US* 


29 


15 50 


14 


5430 


10 


2 


2312 





2 


10 30 


?« 


654 


31 15 50 


14 


55'30 


12 


2 


23 




24 


3 


15 45 


27 


12*54 


34115 5014 


5630 


15 


2 


23 




18 


4 


20 5S 


26 


18,54 


40 


15 51 


14 


57130 


19 


2 


23 




12 


5 
6 


26 13 

31 26 


25 
24 


24 (54 


47 


15 51 


14 


5830 
59)30 


26 

34 


2 


23 




6 



I 0,54 


56 


15 52 


14 


2 


24 


11* 


7 


36 39 


23 


655 


6 


15 53 


15 


1'30 


44 


2 


24 




24 


8 


41 51 


22 


12& 


17 


15 54 


15 


430 


55 


2 


24 




11 


9 


47 22 


21 


18 55 


29 


15 54 


15 


8'31 


9 


2 


24 




12 


10 
11 


52 13 
57 23 


20 
19 


24 55 


42 


15 


15 


12 


31 
31 


23 
40 


3 


25 




6 


2 055 


56 


15 58 


15 


17 


2 


25io 





12 


1 2 31 


18 


656 


12 


15 59 


15 


22 31 


56 


2 


26 




24 


13 


1 7 38 


Vi 


12 56 


29 


16 1 15 


26*32 


17 


2 


27 




18 


14 


1 12 44 


16 


1856 


48 


16 21$ 


3032 


39 


2 


27 




12 


16 
16 


1 17 49 

1 99 *9 


15 
14 

13 


2457 


8 


16 415 


36 


33 
33 


11 

23 


6 


28 




6 


17tl 27 53 


3 057 


30 


16 615 


41 


2 


28 


9 





16 1 32 52 


12 


657 


52 


16 815 


4633 


47 


2 


29 




24 


1911 37 49 


11 


12 58 
1858 
24 58 


12 

31 
49 


16 10 15 
16 11 15 
16 1316 


52'34 
5834 

334 


11 

34 
58 


2 
2 
2 


29 
29 
30 




18 

12 

6 


5 Signs. 
A r orth dene end 




4 • C 
t 


59 
59 


6 
21 


16 14 16 
16 15 16 


9'35 
1435 


22 
45 


2 ' 
2 


30 
31 


8 




24 


\ I Signs. 


South J trend 


12 
18 
24 


59 
59 
60 


35 

48 



16 17 16 


19 36 
2436 
28|36" 





2 


31 
32 
32 




18 

12 

6 


TGis 1 able 
shews the 


16 2C 
lb* 21 


10 
16 


20 
40 


2 
2 




!5 


b0 


11 


16 


3l;37 


O 


2 


32 


7 





Moon's Lati- 


6 


GO 


21 


16 21 16 


32*37 


10 


2 


33 




24 


tude a little 


12 


60 


30116 2216 


37 


[37 


19 


2 


33 




18 


»beyond the ut- 


18 


60 


38 116 22 16 


38 


37 


28 


2 


33 




12 


most limits of 


24 


60 


45 16 2316 


39 


37 


x36 


2 


33 




6 


Ec 


iipses. 




|o 0,60 


45 46 23,10 


39 


U7 


40 U 


331 b 






ARITHMETIC*. 



169 



v 


TABLE XVIII. 


2 he Aioon'* Mean ±,ottgitude } and 


Anumuiy 








for current yearn. 








AD. | Mean Lontr. | Mean Anom. 


^unlromNo e 




Years 


current.| s 


o ' " | s V 


i n 


s o 


i it 






Vol 


7 


1 8 8, 10 12 


34 50 


2 7 


33 33 






1781 


H 


14 42 54 11 22 


19 18 


1 10 


43 18 






1791 


7 


14 54 59 6 5 


39 35 


6 27 


19 46 




B 


1792 





7 28 40 9 17 


26 44 


6 7 


56 52 






1793 


4 


16 51 45 16 


9 59 


5 18 


37 9 






J794 


8 


26 14 51 


3 14 


53 14 


4 29 


17 26 






1795 


1 


5 37 57 


6 13 


36 29 


4 9 


57 43 




B 


1796 


5 


28 11 37 


9 25 


23 38 


3 20 


34 49 






1797 


10 


7 34 43 


24 


6 53 


3 1 


15 6 






1798 


2 


16 57 48 


3 22 


50 8 


3 11 


55 23 






1799 


6 


26 20 54 


6 21 


33 23 


1 22 


35 40 




B 


1800 


11 


5 44 


9 20 


16 38 


1 3 


15 57 






1801 


3 


15 7 5 


18 


59 52 


13 


56 14 






1802 


7 


24 30 11 


3* 17 


43 7 


11 24 


36 31 






1803 





3 53 16 


6 16 


26 22 


11 5 


16 48 




B 


1804 


4 


26 26 57 


9 28 


13 31 


10 15 


53 54 






1805 


9 


5 50 2 


26 


56 46 


9 26 


34 11 






1806 


1 


15 13 8 


3 25 


40 1 


9 7 


14 28 






1807 


5 


24 36 14 


6 24 


23 16 


8 17 


54 45 


B 


1808 


10 


17 9 54 


10 6 


10 25 


7 28 


31 51 


1809 


2 


26 33 


1 4 


53 40 


7 9 


12 8 


.1 


1810 


7 


5 56 5 


4 3 


36 55 


6 19 


52 25 






1811 


11 


15 19 11 


7 2 


20 9 


6 


32 42 




B 


1812 


4 


7 52 52 


10 14 


7 18 


5 11 


9 48 . 


♦ 




1813 


8 


17 15 57 


1 12 


50 33 


4 21 


50 5 






1814 





26 39 3 


4 11 


33 48 


4 2 


30 22 






1815 


5 


6 2 8 


7 10 


17 3 


3 13 


10 39 


i 


B 


1816 


9 


28 35 49 


10 22 


4 12 


2 23 


47 45 




1817 


o 


7 58 55 


1 20 


47 27 


2 4 


28 2 






1818 


6 


17 22 


4 19 


30 42 


1 15 


8 19 






1819 


10 


26 45 6 


7 18 


13 57 


25 


48 36 




B 


1820 


3 


19 18 47 


11 


1 6 


6 


25 42 




, 


1821 


7 


28. 41 54 


1 28 


44 21 


11 17 


5 59 






1841 





12 16 37' 3 8 


28 51 


10 20 


15 44 



170 



LOGARITHMICK 



TABLE 


xix 1 . 


The Hun 


'• A 


.ongitudt for every day 


in the 










year, 


at noon. 










C 

to 


January. 


February. 


March. 


Apri 


i. 1 


s 


May. 

o ' 


s 


June. 


s 


o 


t 


s 


o 


t 


s 


o 


i 


8 


o 


o 

"IT" 


r 
~2 


9 


11 


2110 


12 


5411 


11 


80 


11 


551 


11 12 2 


2 


9 


12 


23*10 


13 


5511 


12 


80 


12 


54'1 


12 10 2 


12 





3 


9 


13 


2410 


14 


5611 


13 


80 


13 


53|l 


13 92 


12 


57 


4 


9 


14 


2510 


15 


57|ll 


14 


8,0 


14 


521 


14 7 


2 


13 


54 


5 


9 


15 


2610 


16 


57|11 


15 


8 







15 
16 


51 

50 


1 


15 5 


2 


14 


52 


6 


9 


16 


27|10 


17 


5811 


16 


8 


1 


16 3 


2 


15 


49 


7 


9 


17 


29' 10 


18 


5911 


17 


80 


17 


491 


17 1 


2 


16 


46 


8 


9 


18 


3010 


20 


Oil 


18 


80 


18 


481 


17 59 


2 


17 


44 


9 


9 


19 


'3110 


21 


Oil 


19 


■ 80 


19 


471 


18 56 


2 


18 


41 


10 
11 


9 


20 


3210 


22 


1 


11 


20 


«!o 


20 


45 


1 


19 54 


2 


19 


39 


9 


21 


3310 


23 


1 


11 


21 


7j0 


21 


44 


1 


20 52 


2 


20 


35 


12 


9 


22 


3410 


24 


211 


22 


7!o 


22 


431 


21 50 


2 


21 


33 


13 


9 


23 


3510 


25 


3'll 


23 


7,0 


23 


41 


1 


22 48 


2 


22 


30 


14 


9 


24 


3610 


26 


311 


24 


60 


24 


40 


1 


23 45 


2 


23 


28 


15 

16 


9 


25 


37(10 


27 


4|11 


25 


60 


25 


39 


1 


24 43 


2 


24 


25 


9 


26 


39.10 


28 


4'ii 


26 


60 


26 


37 


1 


25 41 


2 


25 


22 


17 


9 


27 


3910 


29 


411 


27 


50 


27 


36 


1 


26 39 


2 


26 


19 


18 


9 


28 


4llll 





5J11 


28 


50 


28 


34 


1 


27 36 


2 


27 


17 


19 


9 


29 


42! 11 


1 


511 


29 


40 


29 


33 


1 


28 34 


2 


28 


14 


20 
21 


10 





43J11 


2 


6; 





4'l 





31 


1 


29 32 


2 


29 


11 


10 


1 


4411 


3 


6 





1 


31 


I 


30 


2 


29 


3 





8 


22 


10 


2 


45J11 


4 


6 





2 


31 


2 


28 


2 


1 27 


3 


1 


6 


23 


10 


3 


4611 


5 


7 





3 


21 


3 


26 


2 


2 25 


3 


2 


3 


24 J 10 


4 


4711 


6 


7 





4 


1*1 


4 


25 


2 


3 22 


3 


3 





25|10 


5 


48;li 


7 


7 


5 


1 


1 


5 


23 


2 


4 20 


2 


3 


57 


26I(T 


6 


49 


11 


8 


7 





6 





1 


6 


21 


2 


5 17 


3 


4 


55 


27,10 


7 


50 


11 


9 


8 





6 


59 


1 


7 


20 


2 


6 15 


3 


5 


52 


2810 


8 


51 


11 


10 


8 





7 


59 


1 


8 


18 


2 


7 12 


f? 


6 


49 


29110 


9 


52 











8 


58 


1 


9 


16 


2 


8 10 


o 


7 


46 


3010 


10 


52 











9 


57 


1 


10 


14 


2 


9 7 


3 


8 


43 


3111 


11 


53 











10 


56 








2 


10 5 






. . 



ARITHMETICK. 



171 



TABLE XIX. Concluded. 




July 


. lAugust. 
'so' 


{Sept. 

s o ' 


Octobc 


IV. 
r 


Nov. 


/ 


Dec. 


s 


o 


s 


o 


9 





i 


o ' 


1 


3 


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1 72 



LOGARITHMIC* 



TABLE XX. J /concise 


Equation-Table, adapted to the 


Second Year < 


ifter JLeafi'Yearj and which will be within a 


Minute of the 


Truth for 


every Year ; shewing, to the nearest 


full Minute , how much a 


Clock should be faster or slower 


than the Sun. 


By Mr 


. Smeaton. 


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This Table is near enough the truth for regulating common 
clocks and watches. It may be easily copied by the pen, and 
being doubled, may be put into a pocket-book. 



ARITHMETIC^. 149 

PRECEPTS RELATIVE TO THE PRECEDING 
TABLES. 

To calculate the true time of New or Full Moon. 

Precept 1. Write out the mean time of New Moon in March 
for the proposed year, from Table I. old style, or from Table II. 
in the new ; together with the mean anomalies of the Sun and 
Moon, and the Sun's mean distance from the Moon's ascending 
node. If you want the time of Full Moon in Marchy add the half 
lunation at the foot of Table III. with its anomalies, Sec to the 
former numbers, if the New Moon fulls before the 15th of 
March ; but if it falls after, subtract the half lunation, with 
anomalies, Sec. belonging to it, from the former numbers, and 
write down the respective sums or remainders. 

2. In these additions or subtractions, observe, that 60 seconds 
make a minute, 60 minutes make a degree, 30 degrees make a 
sign, and 12 signs make a circle. When you exceed 12 signs 
in addition, reject 12, and set down the remainder. When the 
number of signs to be subtracted is greater than the number 
you subtract from, add 12 signs to the lesser number, and then 
you will have a remainder to set down. 

In the Tables, signs are marked thus, S degrees thus, ° min- 
utes thus, ' and seconds thus, " 

3. When the required New or Full Moon is in any given 
month after Marchy write out as many lunations, with their 
anomalies, and the Sun's distance from the node, from Table 
III. as the given month is after March ; setting them in order 
below the numbers taken out for March. 

4. Add all these together, and they will give the meantime of 
the required New or Full Moon, with the mean anomalies and 
Sub's mean distance from the ascending node, which are the 
arguments for finding the proper equations. 

5. With the number of days added together, enter Table IV. 
under the given month, and against that number you. have the 
day of mean New or Full Moon in the left-hand column, which 
set before the hours, minutes, and seconds, already found. 

But (as it will sometimes happen) if the said number of days 
fall short of any in the column under the given month, add one 
lunation and its anomalies, &c. (from Table III.) to the foresaid 
sums, and then you will have a new sum of days wherewith to 
enter Table IV. under the given month, where you are sure to 
find it the second time, if the first falls short. 

6. With the signs and degrees of the Sun's anomaly, enter 
Table VII. and therewith take out the annual or first equation 
for reducing the mean to the true syzygy ; taking care to make 
proportions in the table for the odd minutes and seconds of the 

20 



f 



I5# LOGARITHMICK 

iraomaly, as the table gives the equation only to whole degree^ 
Observe, in this and every other case of finding equations, 
that if the signs are at the head of the table, their degrees are 
at the left hand, and are reckoned downwards ; but if the signs 
are at the foot of the table, their degrees are at the right hand, 
and are counted upward ; the equation being in the body of the 
table, under, or over the signs, in a collateral line with the de- 
grees. The titles Add or Subtract at the head or foot of the 
Tables where the signs are found, shew whether the equation 
is to be added to the mean time of New or Full Moon, or sub- 
tracted from it. In the table for reducing the mean to the true 
syzygy, the equation is to be subtracted, if the signs of the 
Sun's anomaly are found at the head of the table ; but it is to be 
added, if the signs are at the foot. 

With the same signs and degrees of the Sun's anomaly, enter 
Table VIII. and take out the equation of the Moon's mean ano- 
maly ; subtract this equation from her mean anomaly, if the 
signs of the Sun's anomaly be at the head of the table, but add 
it if they are at the foot ; the result will be the Moon's equated 
anomaly, with which enter Table IX. and take out the second 
equation for reducing the mean to the true time of New or Full 
Moon ; adding this equation, if the signs of the Moon's anomaly 
are at the head of the table, but subtracting it if they are at the 
foot, and the result will give you the mean time of the required 
New or Full Moon twice equated, which will be sufficiently 
near for common Almanacks. But when you want to calculate 
an eclipse, the following equations must be used : thus, 

8. Subtract the Moon's equated anomaly from the Sun's mean 
anomaly, and with the remainder in signs and degrees, enter 
Table X. and take out the third equation, applying it to the for- 
mer equated time, as the titles. Add or Subtract do direct. 

9. With the Sun's mean distance from the ascending node, 
enter Table XL and take out the equation answering to that 
argument, adding it to* or subtracting it from the former equat- 
ed time, as the titles direct, and the result will give the time of 
New or Full Moon, agreeing with well regulated clocks, or 
watches, very near the truth. But, to make it agree with the 
solar, or apparent time, apply the equation of natural days, found 
in Table XX. and you will have the true time of apparent New 
or Full Moon required. 

The method of calculating the time of any New or Full 
Moon without the limits of the 19th century, will be shown fur- 
ther on. And a few examples with the precepts, will make 
the whole work plain. 

N. B. The Tables begin the day at noon, and reckon for- 
ward from thence to the noon following. Thus, 



ARITHMETIC*. 



,151 



July the 1 3th, at 13 hours (4 minutes 32 seconds of tabular 
time, is July 14th (in common reckoning) at 14 min. 32 sec. 
past 1 1 o'clock in the morning. 



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Jn this example I look for the year 1748 in Table I. O. S. 
against which I find the time of mean New Moon in March, 
Sun's anomaly^ Moon's anomaly, and Sun's distance from Node, 
to be as set down in the example. (Agreeable to Precept I.) I 
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isa XeOGARITHMICK 

Moon, and the Sun's mean distance from the Moon's ascending 
node ; which are the arguments for finding the proper equations, 
(Precept 4.) 

With the signs and degrees of the Sun's anomaly, which, in 
the present case, is signs 24 degrees, I enter Table VII. and 
look for signs at the top of the Table and 24 degrees in the 
left hand column, and find in the angle of meeting, 1 h. 40 m.6 
sec. and by making proportions, in the table (for the odd 59m.45 
sec. (or estimating the Sun's anomaly at 25 deg.) I obtain, for 
the 1st equation, 1 h. 44 m. 1 sec. which I apply to the time of 
mean New Moon, as the title Subtract at the head of the table 
directs. (Precept 6-) 

With the same argument, (namely, 25 deg.) I enter Table 
VIII. and take out thence the equation of the Moons mean ano- 
maly, which in the present instance, I find to be 39 m. 29 sec. 
which I subtract for the Moon's mean anomaly, according to the 
title on the top of the Table. (Precept 7.) 

The result is the argument for finding the 2d equation, with 
which I enter Table IX. and take out as before the next equa- 
tion, applying it to the mean New Moon, as the title directs, 
(precept 7.) This gives the time sufficiently exact for common 
Almanacks. But when you wish to calculate an eclipse, pro- 
ceed according to Precept 8. and 9. 



ARITHMETIC*. 



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To calculate the time of New and Full Moon in a given Year 
and Month of any particular Century between the Christian 
<£ra and 18th Century. 

Precept 1. Find a year of the same number in the 18th Cen- 

. tury with that of the year in the century proposed, and take 

•out the mWi time of New Moon in March old style, for that 

year, with the mean nomalies and Sun's mean distance from the 

node at that time, as already taught. 

2. Take as many complete centuries of years from Table VL 
as, when subtracted from the aforesaid year in the 1 8th. centu- 
ry 1 , ivill answer to the given year ; and take out the first mean 



15$ LOGARITHMICK 

New Moon and its anomalies, &c. belonging to the said centu- 
ries, and set them below those taken out for March in the 18th. 
century. 

3. Subtract the numbers belonging to those centuries, from 
those of the 18th century, and the remainders will be the mean 
time and anomalies, &c. of New Moon in March, in the given 
year of the century proposed. 

Then work in all respects for the true time of New or Full 
Moon, as shown in the above precepts and examples. 

4. If the days annexed to these centuries exceed the num- 
ber of days from the beginning of March taken out in the 18th. 
century, add a lunation and its anomalies, 8c c. from Table III. 
to the time and anomalies of New Moon in March, then pro- 
ceeed in all respects as above. This circumstance happens in 
examples 6* 



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To calculate the true time of New or Full Moon in any giv- 
en Year and Month before the Christian iEra. * 

Precept 1. Find a year in the 18th century, which being ad- 
ded to the given number of years before Christ) diminished by 
1, shall make a number of complete centuries.- 

2. Find this number of centuries in Table VI. and subtract 
the time and anomalies belonging to it from those of the mean 
New Moon in March, the above found year of the V&th century ; 
and the remainder will denote the time and anomalies^ Sou of 



24 



182 



LOGArtlTHMICK 



the mean New Moon in March, the given year before Chris;. 
Then for the true time of that New Moon, in any month of 
that year, proceed in the manner taught before. 



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These Tables are calculated for the meridian of London ; 
but they will serve for any other place, with equal exactness, 
by subtracting 4 minutes from the tabular time, for every de- 
gree that the meridian of the given place is westward of Lon- 
don, or adding 4 minutes for every degree that the meridian oC 
the given place is eastward of London; as inexample VII. 



ARITHMETIC*. 




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trie's o 2 

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To calculate the true time of New or Full Moon in any giv- 
en year after the 19th, Century. 

Precept 1. Find a year of the same number in the 18th. 
century with that of the year proposed, and takeout the time 
and anomalies, &c. of New Moon in March, old style, for that 
year, in Table I. 

2. Take so many years from Table VI. as, when added to 
the above-mentioned year in the 1 8th, century, will answer tc* 
the given year in which the New or Full Moon is required ; and 
take out the first New Moon, with its anomalies, for these, con*- 
pletc centuries. ' 



m 



LOGARITHMICK 



3* Add all these together, and then work in all respects a? 
shewn above, only remember to subtract a lunation and its ano* 
roalies, when the above mentioned addition carries the New 
M?on beytmd the 3 1st of March ; as in the following example * 



48 

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In keeping by the old style, we are always feure to be right, by 
adding or subtracting whole hundreds of years to or frpm any 
given year in the 18th century. 

But in the new style we may be very liable to make mistajs^, 
on account of the leap-year's not coming in regularly 'every 



ARITHMETICS 



I8S 



fourth year : And therefore, when we go without the limits of 
the 18th century, we had best keep to the old style, and at the 
end of the calculation reduce the time to the new. '• Thus, in the 
22d century there will be 14 days difference between the styles; 
and therefore, the true time of New Moon* in this last exam- 
ple being reduced to the new style, will be the 22 d of Jujy, at 
37 in. 33 sec. past 1 P. M. 

To find the times of mean New and Full Moon for ever^ 
month in the year; 

Precept %. Find the mean time of New and Full Moon in 
the month of January, as already taught, and to these times add 
3 mean lunation, viz, 29 D. 12 H. 44 M. 3 S. continually, rejec- 
ting the days in the month wherein the mean New or Full 
Moon is required^ and the results will denote the times re- 
quired. 

EXAMPLE, 

Let it be required to find the mean New and Full Moons in 
every month of the year 1797. 



Mean Full Moons. 

Month*, po nor 



Mean New Moons. 



January 


12 


13 18 18 


} Lunation + 


29 


12 44 ^ 


February 


11 


2 2 21 


March 


12 


14 46 24 


April 


11 


3 30 27 


May 


10 


16 14 30 


June 


9 


4 58 33 


July 


8 


17 42 36 


August 


7 


6 26 39 


September 


5 


19 10 42 


October 


5 


7 54 45 


November 


3 


20 38 48 


Pecember 


3 


9 22 51 



' Months. 


| D. H. 


January 


27 7 


1 Lunation 


+ 29 12 


February 


25 20 


March 


27 9 


April 


25 21 


May 


25 10 


June 


23 23 


July 


23 12 


August 


22 


September 


20 13 


October 


20 2 


November 


18 15 


December 


18 3 



M 1 S. 
40 21 
44 3 
24 24 

8 27 
52 30 
36 33 
20 36 

4 39 
48 42 
32 45 
16 48 

S\ 
44 54 



PERMISCUO US Q UESTIOJVS* • 

1 . Required the true mean time of New Moon in May y old style* 
the year before Christ 585, and the Sun's distance, at that time, 
i'rom the Moon's ascending node ? 

Ans. True New Moon was on the 28th day, at 2 m. 31 sec., 
past IV in the evening; and the Sun's distance from node was 
3 deg. 50 m. 47 sec. (Sun eclipsed.) 

2. Required the time of tiue New Moon, and Sun's distanAf 
from the /iode, tn July !890 A. D. old style ? 

■/ 



*16 LOGARITHMIC* 

Ana. N. M. 29tU day III o'clock 52 m. 28 sec. and Sun's disti 
from node 1 1 S. 29 ° 56' 57 ". (Sun eclipsed.*) 

3. Required the true time of New Moon, and at that time the 
Sun's distance from the ascending node, in April, A. D. 1819, 
N. S. Ans. New Moon 24th day at 46 m.21 sec. past XI. A. M. 
and Sun's dist. from node 12 ° 10 ' 9 ". (Sun eclipsed, visible.) 

4. Required to find the true time of opposition of the Sun and 
Moon, and the simultaneous distance of the Sun from the node, 
in October, A. D. 1819, N. S. 

Ans. True Full Moon 3d day, at 7m. 48 sec. past III. in the 
evening, and Sun's dist* from descending node 50 ' 5 " (Moon 
eclipsed, total.) 

5. Required the true time of conjunction of the Sun and 
Moon, September, A. D. 1820, N. S. 

Ans. On the 7th day, at 16 m. 55 sec. past II. P. M. (Sun 
eclipsed.) 

6. Required the true time of Full Moon at Boston, Long. 70 * 
37 ' 15 " W. in May, A. D. 1826, N. S. 

Ans. 21st day, at 29 m. 58 sec. past X in the morning. (Moon 
eclipsed.) 

7. Let it be required to find the true time of New Moon m 
July, 1980, O. S. and how far short the Sun will be at that time 
from the Moon's ascending node. 

^ Ans. 29th day, at 52 m. 28 sec. past III. in the morning. 
And the Sun will be only 3 *, 3 ", short of the Moon's node, N. 
ascending. (Consequently, the Sun must suffer a total eclipse.) 



To calculate the true Place of the Sun for any given Moment of 

Time. 
Precept 1. In Tabic XII. find the next lesser year in num- 
ber to that in which the Sun's place is sought, and write out 
his mean longitude and anomaly answering thereto : to which 
add his mean motion and anomaly for the complete residue of 
years, months, days, hours, minutes, and seconds, down to the 
given time, and this will be the Sun's mean place and anomaly 



* Note. When the 9uh is within 17 degrees of either of the 
Moon's nodes at the time of New Moon, he will be eclipsed at 
that time: and when he is within 12 degrees of either of the 
nodes at the time of Full Moon, the Moon will be eclipsed a* 
jj^at time.— See the method of calculating Eclipses. 



ARITHMETICS 



J W 



at that tune, in the old style ;* provided the said time be in 
anyfyear after the Christian -£ra. 

2. Enter Table XIII. with the Sun's mean anomaly ; and ma- 
king proportions for ttte odd minutes and seconds thereof, take 
out the equation of the Sun's centre : which being applied to his 
mean place, as the title Add or Subtract directs, will give his 
true place or longitude from the vernal equinox, at the time for 
which h was required. 

EXAMPLE I. 

Required the Sun's true place, July 13th 1748, Old Style, at 
23 hours 19 minutes 58 seconds past noon ? In common reck- 
on in % t July 14th, at 19 minutes 58 seconds past XI. in the fore* 
noon. 



To the radical year after 
Christ iroi 

Add complete years 



Bissextile, Days 
Hours 
Minutes 
Seconds. 



C 40 

I 7 

July 
13 
23 
19 
58 



Sun's mean place at the given time. 
EquationoftheSun'scentre,subtract 

Sun's true place at the same time. 



Sui 


i*s Long. 


| Sun's Anom. 


S 


' 


t» 


S ' '»» 


9 


20 43 


SO 


6 13 1 





18 


8 


11 29 37 


11 


29 18 


49 


P. 29 11 


5 


28 24 


8 


5 28 24 





13 47 


57 


13 47 57 





56 


40 


56 40 








46 


46 








2 


2 


4 


3 30 


20 


24 58 25 




— 47 


57 


Mean Anom. 


4 


2 42 


23 


or 2 42 26 



of Cancer. 



* N. B. Although this Table is constructed according to the 
t>ld Style, yet it will serve, with equal exactness, for the New, 
T>y diminishing the day of the month in this Table by 12, for the 
present age. Thus, suppose the required time to be on the 
28 th day of May N. S. Instead of the numbers answering to that 
day, write out those for the 16th day, Sec. But if the required 
time be within the limits of the 18th century, subtract 11 days 
from the given time. See Exmnflle II, on the next page. 



1 



Jaa 



LOGARITHMIC^ 



EXAMPLE II. 
* Required the Sun's true place, March A 1st, 1764, jfew Stytej 
at 22 hours 30 minutes 25 seconds, ffcast the noon of that day ? 



Add complete years 



To the radical year after 
Christ 1701 

;6o 

3 

March 

3 1 st day—- 1 1 = Bissextile,Day s 20 

Hours 22 

Minutes 30 

Seconds 25 

Sun's mean place at the given time. 
Equations of Sun's centre, add 

Sun's true place at the same time. 



Sun's Long. | Sun's Anom. 



S 



S 



9 20 43 S0\ 6 13 1 O 
27 12 11 29 26 
II 29 17 11 29 14 Q 



1 28 9 11 

20 41 55 

54 13 

1 14 

1 



10 14 36 
-h l 55 31 



1 28 9 

20 41 5$ 

54 13 

1 14 

1 



9 1 27 23 
Mean Anom. 



12 10 



7J or 12 10 
of Aries. 



EX. III. Required the Suns/ true place and anomaly, July 
28th, 15h. 52m. 26 sec. past noon, in the year 1980, 9. S. 

Ans. *S. 18° 35 '8 "from the vernal equinox, and Sun$ 
anomaly, 15.7° 18' 9"- 



Tofirtd the Sun** Distance from the Moon* 9 Ascending JYode> 
at the time of any given JVew or Full Moon ; and consequently 
to know whether there is an Eclipse at that Time, or not. 

The Sun's distance from the Moon's ascending node, is the 
argument for finding the Moon's fourth equation in the syzy- 
gies, and therefore it is taken into all the foregoing examples 
in finding the times of these phenomina. 

Thus, at the mean time of New Moon in July, 1748, the Sun's 
mean distance from the ascending node is 5 S. 25 ° JO ' 1 ". See 
Example /.'page 175. 

The descending node is opposite to the ascending one,and they 
are, therefore, just six signs distant from each other. 

When the Sun is within 17 degrees of either of the nodes at* 
the time of New Moon, he will he eclipsed at that time : and 
when he is within 12 degrees of either of the nodes at the time 



ARITHMETICS.. M 

of Full Moon, the Moon will be eclipsed.* Thus we find these 
-will be an eclipse of the Sua at the time of New Moon in July, 
1748. 

But the true time of that New Moon comes out by the equa- 
tions to be £ minutes 10 seconds later than the mean time 
thereof, by comparing these times in the above example : and 
therefore, (in this, and all similar cases) we must add the Sun's 
motion from the node during that interval to the above mean 
distance, 5 S. 25 ° 30 » 1 '% which motion is found in Table XII 
for 6 minutes, 10 seconds, to be 14 ". And to this we must ap- 
ply the equation of the Sun's mean distance from the node, in 
Table XV. found by the Sun's anomaly, which at the mean time 
of New Moen in example I. we estimated at 25 °, and then we 
shall have the Sun'* true distance from the node, at the true 
time of New Moon, as follows ; 

Sun from Node, 

sO'" 

At the mean time of New Moon? e , ! « A , 

in July, 1748, £ 5 25 30 1 

Sun's motion from the > 6 minutes \ 1 4 

node for J 10 seconds. J O 

Sun's mean distance from node,? _ - «. n ,- 

at true New Moon $ 5 **• oU 15 

Equation of mean distance from ? -« 

node, subtract } ♦ 

Sun's true distance from < 5 25 29 23 

the ascending node ; that is, { 4 • 30 ' 37 **■ from 

the descending node ; which being far within the above limit 
of 17 degrees, shows that the Sun must then be eclipsed.. 

And now we shall shew how to project this, or any other 
eclipse, either of the Sun or Moon. 



* Note. This admits of some variation : for inapogeal eclipses 4 
the solar limit is but 1 6\ degrees ; and in perigcal eclipses, it 
is 18}. When the Full Moon is in her apogee, she will be eclip- 
sed if she be within lOf degrees of the node ; and when she is 
full in her perigee, she will be eclipsed if sh$ be within 12^. 
degrees of the node. 

25 



if* LOGARITHMIC* 



TO PROJECT AN ECLIPSE OF THE SUN. 

In order to this, we must find the ten following Elements by 
means of the Tables. 

1. The true time of conjunction of the Sun and Moon; and 
at that time, 

2. The semidianieter of the Earth's disk* as seen from 
the Moon, which is always equal to the Moon's horizontal par* 
*llax. 

3. The Sun's distance from the solstitial colure to which he 
is then nearest. 

4. The Sun's declination. 

5. The angle of the Moon's visible path with the ecliptick. 

6. The Moon's latitude. 

7. The Moon's true horary motion from the Sun. 

8. The Sun's semidianieter. 

9. The Moon's ,semidiameter. 

10. The semidianieter of the penumbra. 
Wc shall now proceed to find these elements for the Sun's 
Eclipse in July, 1748, O. S. 

1. To find the true time of JYew Moon. This, by example I. 
page 175, is found to be on the 14th dav of the said month, at 19 
minutes 58 seconds past XL in the morning. 

2. To find the Moon 9 8 horizontal parallax ', or semidiameter of 
the Earth'* disky as seen from the Moon. Enter Table XVIL 
with the signs and degrees of the Moon's anomaly, (making 
proportions, because the anomaly is in the table only to every 
6th degree,) and thereby take out the Moon's horizontal paral* 
lax ; which, for the above time answering to the anomaly 10 ° 
56 '56", is 54* 33". 

3. To find the Sun's distance from the neartst solstice, viz. 
the beginning of Cancer, which is 3 signs } or 90 degrees fromthe 
beginning of Aries. 

It appears by the example on page 187 (where the Sun's 
place is calculated to the above time of New Moon) that the 



*. Note. The body, or face of the Sun, or Moon, as it appears, 
to a spectator on the Earth j or of the Earth, as it would appear 
to a spectator at the Sun, or Moon, is called its Disk. 



ARITHMETICS. in 

Sun'fi longitude from the beginning of Aries is there 4S. 2° 42* 
33", that is 2° 42* 23" from the beginning of Cancer ; Thus 

SO'" 

From the Sun's Longitude or place 4 2 42 23 

Subtract, . ■ 3 



Remains the Sun's distance from the > 1 2 42 23 
solstice of Cancer. J 

Or, 32° 42* 23" : each sign containing 30 degrees. 

4. To find the Sun'* declination. Enter Table XIV. with the 
signs and degrees of the Sun's true place, viz. 4£, 2° and mak- 
ing proportion for the 42' 23", take out the Sun's declination 
answering to his true place, and it will be found to be 19° 38' 8" 
North. 

5. To find the Moon's latitude. This depends on her distance 
from her ascending^ node, which is the same as the Sun's dis- 
tance from it at the time of New Moon : and with this the 
Moon's, latitude is found in Table XVI. 

Now we have already found that the Sun's equated distance 
from the ascending node, at the time of New Moon in July 
1748, is 5$. 25° 29* 23". See the 189th fiage. Therefore, 
enter Table XVI. with 5 signs at the bottom, and 25 and 26 de- 
grees at the right hand counted upward, and take out 26* 13', 
the latitude for SS. 25° ; and 20 e 59', the latitude for 5&. 26' : 
and by making proportion between these latitudes for the 29* 
23" by which the Moon's distance exceeds the 25th. degree; 
her true latitude will be found to be 23 ' 56 " North Ascending. 

6. To find the Moon's true horary motion from the Sun. With 
thfe Moon's anomaly, viz. OS. 10*56' 56", enter Table XVII. 
*nd take out the Moon's horary motion ; which, by making 
proportion in that table, will be found to be 30' 14". Then, 
with the Sun's ajnomaly, 25°, take out his horary motion 2' 23" 
from the same table ; and subtracting the latter from the for- 
mer, there will remain 27' 51" for the Moon's true horary mo* 
tion from the Sun. 

7. To find the angle of the Moon's visible fiath with the Eclifi- 
tick. This, in the projection of eclipses, may be always rated 
at 5° 55', without any sensible errour. 

8,9. To find the semi diameters of the Sun aud Moon. These 
are found in the same Table, and by the same arguments, as 
their horary motions.— In the present case the Sun's anomaly 
gives his semidiameter 15' 51", and the Moon's anomaly gives 
her diameter 14' 56'*. 

10. To find the semidiameter of the Penumbra. Add the 
Moon's semidiameter to the Sun's, and their sum will be the 
serajdiarneter of the penumbra, viz. 30' 47". 



o 


» 


»» 





54 


33 


32 


42 


23 


19 


38 


8 





23 


36 


37 51 


5 


35 


a 




15 


51 




14 


56 




30 4T 



t» LOGARITHMIC* 

Now collect these elements, that they may be found the mom 
readily when they are wanted in the construction of this Eclipse, 

D. H. M. S. 
1. True time of New Moon in July, 1748, 14 11 19 58 



2. Sun's diameter of Earth's disk, 

3. Sun's distance from the nearest solstice, 

4. Sun** declination, North, 

5. Moon's latitude North descending, 

6. Moon's horary motion from the Sun, 

7. Angle of the Moon's visible path with 

the ecliptick, 

8. Sun's semidiameter, 

9. Moon's semidiameter, 

40r Semidiameter of the penumbra, 



TO PROJECT AM ECLIPSE OF THE SUM 
GEO ME TRICALL F. 

Make a scale ef any convenient length, as A. C. (Fig. 1.) and 
divide it into 60 equal parts, reckoning each part to be one 
minute, or the sixtieth part of a degree. 

Then, take the semidiameter of the Earth's disk, 54 minutes, 
33 seconds, (or 54$) from the scale, in your compasses ; and with 
that extent, set one foot in the end C of the scale as a centre $ 
and with the other foot describe the semicircle A D B, for the 
circumference of the northern half of the Earth's illuminated 
disk, or surface, because we live on the north side of the Equa- 
tor ; continue the line A C to B ; so A C B shall be a portion 
of the Ecliptick, equal to the diameter of the Earth, as seen from 
the Sun, or Moon at that time. 

Upon the centre C, raise the straight line C D H, perpendic- 
ular to A C B ; and call the line C D H, the axis of the ecliptick. 

Being provided with a good sector, open it to the radius C A 
in the line of chords; and taking from thence the chord of 23$ 
degrees in your compass, set it off both ways from D to G and 
to E, in the periphery of the semi-disk. [But, as much the 
greater number of those into whose hands this work may faJL, 
are not supposed to be thoroughly skilled in the use of Mathe- 
matical Instruments, we shall pursue somewhat a different 
method ; which, in point of simplicity and precision, is no less 
preferable :] Or :— 

Divide the quadrants A D and D B, each into 90 equal pajts, 



ARITHMETICS. l« 

ifcr degrees, beginning at D. Then connect the points E and Gr 
(which are distant 2 3£ degrees on each side of D) with the 
straight line E F G ; in which the North pole P of the Earth's 
disk will always be found. 

When the Sun is in Aries, Taurus, Gemini, Cancer, Leo, and 
Virgo, the North pole of the Earth is enlightened by the Sun ? 
but while the Sun is in the other six signs, the South pole i» 
enlightened, and the North pole is in the dark. 

And when the Sun is in Capricorn, Aquarius, Pisces, Aries, 
Taurus, and Gemini ; the northern half of the Earth's axis 
C XII P lies to the right hand of the axis of the ecliptick, as 
seen from the Sun ; and to the left hand, while the Sun is in the 
other six signs. 

The order, and the names of the Signs, the months and days 
of the year, in which the Sun appears to enter these Signs, are 
as follows : 



J l > 


(2-) 


(30 


<*•) 


(«•) 


(60 


Arte a f 


Taurus, 


Gemini, 


Cancer, 


Leo, 


Virgo, 


March, 


April, 


May, 


June, 


July, 


August, 


20, 


20, 


31, 


21, 


23, 


23, 



(*•) (•■) (9.) Co,) (.11,): (12,) 

Libra, ocorflio, Sagittarius, Cafiricornus, Aquarius, JPiscct, 
Sept. October, November, December, January, February* 
23, 23, 22, 21 20, 19. 

Open the sector, till the radius (or distance of the two 90*s) 
of the signs be equal to the length of D G, and take the sine of 
the Sun's distance from the solstice (32 8 42 * 23 ") as nearly 
as you can guess, in your compasses, from the line of sines, and 
set off that distance from F P, in the line E F Q, because the 
Earth's axis lies to the left hand of the axis of the ecliptick, as 
«een from the Sun in the month of July. Or ; 

Set one foot of the compasses in the point F, where the line 
E F G intersects the axis of the ecliptick CDH; and, having 
extended the other foot from F to E, or from F to G, describe 
the semicircle E H G, and divide its quadrant H E into 90 equal 
parts or degrees. — If the Earth's axis had lain to the right hand 
from the axis of the ecliptick, the quadrant H G must have 
been divided into 90 degrees, and not the quadrant HE. 

As the Sun is 32 degrees 42 minutes 23 seconds, (which 
may be estimated 3 2 -J- degrees) from the nearest (or summer) 
solstice, which is the lirst point of Cancer, on the noon of the 
14th of July 1748, draw the right line I P, parallel to H D, from 
32| degrees of the quadrant H E till it meets the line E F G at 



*9* LOGARITHMIC* 

P, then from P to C, draw the right line PC; so P C shall he 
the northern half of the Earth'* axis, and P the North, pole. 

As the Sun is on the North side of the Equator in, July , and 
Consequently nearer the point of the heaven just over London 
(or the vertex of London) than the Equator is ; subtract his de- 
clination, 19 degrees 38 minutes (neglecting the 8 seconds) 
from the Latitude of London, 51 degrees 30 minutes, and the 
remainder will be 31 degrees 52 minutes, for the Sun's distance 
from the vertex of London on the noon of July the 14th. 

From the point k (in the right hand side of the semicircle 
ADB)at 31 degrees 52 minutes, counted upward from B, 
draw k I, parallel to C D : and taking the extent* / in your 
compasses, set it from C to XII on the Earth's axis C P. So, 
the point XII shall be the place of London, at the instant when 
it is noon at that place on the 14th. of July 1748. 

Add the Sun's declination 19° 38% to the Latitude of London 
51° 30*, and the sum will be 71 degrees 8 minutes, for the Sun's 
distance from the vertex of London on the 14th of July at mid- 
night. Therefore, 

From 71° £', counted upward from B to m in the right hand 
side of the semicircle A D B, draw the right line m n parallel 
to C D. 

Then, taking the extent m n in your compasses, set it from 
C towards or beyond P on the Earth's axis C P, as it happens 
to reach short of P or beyond it : but in the present case, it 
reaches so little above P, that we may reckon C P, lo be its 
whole extent : and so, the point P shall lepresent the place or 
situation of London at midnight, beyond the illuminated part of 
the Earth's disk, as seen from the Sun ; and consequently, in 
the dark part thereof! 

Divide the part of the Earth's axis between XII and P into 
two equal parts, XII X and P K ; then, through the* point K, 
draw the right line VI K VI (till it meets, on each side, the 
periphery of the disk) perpendicular to the Earth's axis C XII 
K P. 

Now, to draw the parallel of latitude of any given place, as 
suppose London, or the path of that place on the Earth's en* 
lightened disk, as seen from the Sun, from Sun-rise till Sun-set, 
proceed as follows* 

Subtract the Latitude of London, 5l p 30% from £0° 00', and 
there will remain 38| for its colatitude, which take in your com* 
paste's, from the line of chords, making C A or CB radius 5 Or, 

From 38$ degrees, counted upward from B to y in the semi- 
circle A D B, draw the right line v w ; and, having taken its- 
length in your compasss, set off that extent both ways from K in 
the Earth's axis, to VI and VI, in the line VI K VI.. ; -, 



A*UT»MET1CR. tfji 

The compasses being opened from K to VI, set one foot in K as 
a centre } and with the other describe the semicircle VI 7 8 9 10 
i 1 12 1 2 3 4 5 VI, and divide it into 12 equal parts, Then, 
from these points of division (7 8 9, Sec.) draw the dotted lines 

7 a, 8 by 9 c, 10 d 9 Sec. all parallel to the Earth's axis C XII P, 
as in the figure. 

With the small extent P K as radius, describe the semicir- 
cle P 6 5 4 3 2 1 XII, and divide the lower quadrant into 6 equal 
parts as in the points 1,2, 3,4,5, 6 ; because the Sun has 
North declination. 

But if the Sub had South declination, the other quadrant 
must have been so divided. 

Through the said division points of the quadrant XII 1 2 S 4 
Ice. draw the rightlinesXI 1 XI, X 2 X, IX S IX, VIII 4 VIII 
VII 5 VII, all parallel to the right line VI K VI ; and through 
the points where these lines meet the former parallel lines 7 a, 

8 by 9 c, 10 dy &c draw the elliptical curve VI VII VIII IX X 
XI XII I II III IV V VI, which may be done by hand, from 
point to point ; and set the hour-letters to these points where 

the right lines meet in the curve, as in the figure.* 

This curve shall represent the parallel of Latitude of London, 
or, the path which London (by the Earth's motion on its axis) 
appears to describe on the Earth's disk, as seen from the Sun on 
the 14th of Julyy from VI in the morning untill VI in the eve* 
ning : and the points VI, VII, VIII, IX, Sec. in the curve shall 
be the point of the disk where London would be at each of these 
hours respectively, as seen from the Sun. 

If the Sun's declination had been as far South as it was North, 
the diurnal path of London would have been on the upper side 
of the line VI K VI ; that is the ellipse, of which the curve VI 
VII, VIII, IX, X, &c. is a part, would have been complete? 
and must have been regulated by dividing the upper quad- 
rant P 6 (of the small semicircle) into 6 equal parts, and 
drawing lines parallel to VI K VI, as before, till they meet 
the intercepting lines drawn through the division points of the 
quadrant P C. The points in which this elliptical curve would 
touch the periphery of Earth's disk, would denote the instant 
of the Sun's rising, and of setting at the given place. 

MakeC AorCB the radius of a line of chords on the sec* 
tor, and take therefrom the chord of 5 ° 35 ', the angle of the 
Moon's visible path with the Ecliptick : Or, 

From the point M, viz. at 5 degrees 35 minutes, to the right 

*N. B. The hour letters on the right hand side of XII, to- 
wards VI (in the Figure) viz. XI X IX VIII VII, is an errour 
in sculpture ; it ought to be I ll III IV V Sec. The reader is 
therefore required} to correct this trivial mistake ki projecting 
Eclipses. 



1,96 LOGAMTHMICK 

hand of the axis of the Ecliptick C Dy draw the right line 
M C, for the axis of the Moon's orbit as seen from the Sun, be- 
cause the Moon's Latitude is North de acending % on the 14th/uty 
1748. If her Latitude had been North ascending^ the axis of 
her orbit must have been drawn 5 degrees 35 minutes on the 
left hand from the axis of the Ecliptick. . 

N. B. The axis of the Moon's orbit lies the same way when 
her Latitude is South-ascending, as when it is North-ascend* 
ing ; and the same way when South descending, as when North 
descending. 

Take the Moon's. Latitude 23' 36", from the scale C A in 
your compasses, and set that extent from C to q on the axis* 
(C D) of the Ecliptick. Then, through the point y, draw the 
right line IX X XI q z XII 1 &c. perpendicular to the axis of 
the Moon's orbit C z M ; and this line shall be the path of the 
centre of the Moon's shadow over the Earth r and will repre- 
sent as much of the Moon's orbit, seen from the Sun, as she 
moves through, during the time that her shadow or penumbra is 
passing over the Earth- 

. From C, on the scale AC, take the Moon's horary motioa 
from the Sun, 27* 51 ", in your compasses ; and make the 
small scale A B (Fig. 2.) equal in length to that extent .• and 
divide this scale into 60 equal parts, for so many minutes of 
time. Then, as the time of New Moon, on the 14th of July, 
1748, was 19 minutes, 58 seconds, after XI o'clock, take 19 
minutes, 58 seconds, or, in this case, 20 minutes, (not regard* 
ing the 2 seconds) counted from A to « en the small scale A B 
in your compasses, and set them off, (in Fig. !.) from the middle 
point between q and £, in the path of the penumbra's centre, to • 
XI in that line ; because the instant of tabular time of New 
Moon is exactly between the point q, where the axis C D of the 
Ecliptick, and the axis C M of the Moon's orbit, intersect the 
line, or path of the penumbra's centre on the Earth. 

Take the whole length of the scale A B (Fig. 2.) in your com- 
passes ; and with that extent, make marks along the line IX X 
XI XII I, Sec both ways from XI ; and set the hour-letters to 
these marks, as in the figure. Then, from the scale A B (Fig.2.) 
divide each space, from mark to mark, into 60 equal parts, or> 
horary minutes, which shall shew the points of the Earth's disk 
where the centre of the penumbra falls, at every hour and min- 
ute, 1 during its transit over the Earth. 



[To the Binder.— JLet the Plate face this page, and unfold to 
the right.] 



cttheSuxb&ty* 



observed 



Afnr**"*""* pis, Old StOr. 






^o 



A 



itr-. 



_30 



ARITHMETICS <y>7 

Apply one side of a square* to the line of the penumbra's 
path, IX X XI, &c. and move the square forward and back* 
ward till the other side cuts the same hour and minute, as at 
r and «, both in|the path of the penumbra's centre, and the path of 
London : and the minute which the square cuts at the same 
instant in both, these paths, is the ins tan c of the visible conjunc- 
tion of the Sun and Moon, or the greatest obscuration of the 
Sun, at the place for which the construction is made, namely, 
Lx>ndon f in the present example ; and this instant, according to 
the projection, is at 34£ minutes past X o'clock in the morning; 

Take the Sun's semidiameter 15 '51 "in your compasses, j 

from the scale A C, (Fig. 1 .) j and setting one foot at r, as a cen- 
tre in the path of London, namely, at 34J minutes past X, with < 
the other foot describe the circle R S for the Sun, or which shall ; 
represent the Sun's disk as seen from London, at the greatest j 
_ obscuration. \ 
Then take the Moon's semidiameter 14' 56" in your com- 
passes from the scale ; and setting one foot in the Moon's path ' -\ 
at *, 34* minutes past X, with the other foot describe the circle < 
T U, for the Moon's disk, as seen from London, at the moment 
when the eclipse is at the greatest ; and the portion of the Sun's 
disk which is hid or obscured by the Moon's, will shew the 
quantity of the eclipse at that time ; which quantity may be * 
measured on a line as, 1 2 3 4 5 6, Sec. equal to the Sun's diam* 
eter, and divided into 12 equal parts for digits ;t of which ac- 
cording to the present projection, there are 9f digits eclipsed. 
Lastly, take the semidiameter of the penumbra 30' 47" from 
the scale C A, (fig I.) in your compasses ; and setting one foot 
in the path of the penumbra's centre, direct the other foot t© 
the path of London among the morning hours at the left hand ; 
and carry that extent backward and forward, till both points of 
the compasses fall into the same instant in both the path's ; and 
that instant will denote the time when the Eclipse began at 
London. Then, do the like on the right hand of the axis of 
the ecliptick ; and where the points of the compasses fall into 
the same instant in both the paths, that instant will be the time 
when the Eclipse ended at London. « 
These trials give 7\ minutes after IX in the morning for the 
the beginning of the Eclipse : 34£ minutes after X, for the time 
of greatest obscuration ; and 13* past XII, for the time when 
the Eclipse ended. 



Note. *The learner will find it convenient to be provided 
with a small wooden square , the two sides of which are about 
6 inches in length* 

t A Digit is a 12th part of the apparent diameter of the 
Sun or Moon. 

26 



m L0GARITHM1CK 

From these times we must subtract the equation of natural 
days, viz. 6 minutes, in July 14th. and we shall have the appar- 
ent times ; nanaeJy, I minute SO seconds past IX, for the be- 
ginning of the Eclipse, 28 minutes SO seconds past X, for the 
time of greatest obscuration, and 7£ minutes past XII for the 
time when the Eclipse ended. But the most convenient way is 
to apply this equation to the true equal time of New Moon, 
before the projection be begun, as is done in Example I. 

For the motion or position of places on the Earth's disk, an 
swer to apparent or solar time.— f See Mr. Patterson's Edition 
of Fuge son's Astronomy % page 340,— also his introduction t& 
Agronomy y p. 163—171, and End. Mat Phil, fi. 391—394. 



TO PROJECT AM ECLIPSE OF THE M00M. 

When the Moon is within 12 degrees of either of her Nodea^ 
at the time when she is full, she will be eclipsed, otherwise she 
Will not. 

We find by Example III. page 178, that at tlie time of mean 
Full Moon in April) the Sun's distance from the ascending node 
was 1 IS 26° 53* 2" ; that is only 3* 6' 58" short of her descen- 
ding node, and the Moon being then Opposite to the Sun, must 
have been just as near her ascending node, and was therefore 
eclipsed. 

The Elements for constructing an Eclipse of the Moon are 
eight in number, as follows : 

1. The true time of Full Moon .• and at that time. 

2. The Moon's horizontal parallax, 

3. The Sun's semidiameter. 

4. The Moon's semidiameter. 

5. The semidiameter of the Earth's shadow at the Moon. 
6u The Moon's Latitude. 

7* The angle of the Moon's visible path with the Ecliptick. 

3.. The Moon's true horary motion from the Sun. 
Therefore, 
* 1. To find the true time of Full Moon. Work as already 
taught in the Precepts. — Thus we have the true time of Full 
Moon iu Aprils 1819, (see Example II L page J 78, J on the 
10th day, at 1 1 minutes 48 seconds past I o'clock, P. M. 

2. To find the Moon's horizontal parallax. Enter Table 
XVII. with the Moon's mean anomaly (at the above time of Full 
Moon) 2 5 27 ° 43 ' 47 ", and thereby take out her horizontal 
parallax ; which, by making the requisite proportion, will be 
found to be 57 • 20 ". 



ARITHMETICS W 

3, 4. To find the aemidiameter tf the Sun and Moon. Enter 
Table XVII. with their respective anomalies, the Sun's being 
9 5 7 ° 38 ' 53 ", (by the aforesaid Example) and the Moon*s 2 
S 27 ° 43 " 47 " ; and thereby take out their respective semidi- 
ameters : The Sun's 16 ' 4 ", and the Moon's 15 ' 38 ". 

5. To find the a e mi diameter of the Earth* 8 shadow at the 
Moon. Add the Sun's horizontal parallax, which is always 10 ", 
to the Moon's, which, in the present case is 57 ' 20 ", and the 
sum will be 57 * 30 ", from which subtract the Sun's semidiame- 
ter 1 6 ' 4 ", and there will remain 41 * 26 M for that part of t|te 
Earth's shadow which the Moon then passes through. 

• 1. To find the Moon* a Latitude, Find the Sun's true dis- 
tance from the ascending node (as already taught in page 189) 
at the true time of Full Moon ; and this distance, increased by 
six signs, will betfte Moon's true distance from the same node^ 
and consequently the argument for fiindingher true latitude, as 
shewn in page 191. 

Thus, in Example III. the Sun's mean distance from the as- 
cending node, was 115 26 ° 53 ' 2 ", at the time of mean Full 
Moon : but it appears by the Example, that the true time there- 
of, was 13 hours, 55 minutes, 33 seconds, later than the mean 
time, and therefore we must add the Sun's motion from the 
node (found in Table XII.) during this interval, to the above 
mean distance 1 13. 26° 53' 2", in order to have his mean dis- 
tance from it at the true time of Full Moon. Then to this apply the 
equation of his mean distance from the node (found in Table 
XV.) by his mean anomaly 95 7° 38' 53" ; and lastly, add si* 
signs s so shall the Moon's true distance from the ascending 
node be found, as follows : 

s ' M 

Sun's distance from node at mean Full Moon, II 26 53 2 

f 13 hours 32 2 

Add his motion from it in< 55 minutes 2 15 

1 33 seconds, 1 



11 


27 


27 


20 




2 


4 





11 


29 


31 


20 


6 












Sun's mean distance at true Full Moon* 
Equatipn of his mean distance, add 

Sun's true distance from the node. 
To which, add 

And the sum will be 5 29 31 20 

Which is the Moon's true distance from her ascending node 
at the true time of her being full ; and consequently the argu- 
ment for finding her true latitude at that time. Therefore, 
with this argument, enter Table XVI. making proportion, 
trelwcen the latitudes belonging to the 5th and 6th degree of 



200 LOGARITHMICK 

the argument at the right hand (the signs being at the bottom) 
for the 31* 20", and it will give 2' 41" for the Moon's true Lat- 
itude, which appears by the Table to be North descending. 

7. To find the angle of the Moon 9 a visible path with the Eclifi-' 
tick. This may be reckoned 5° 35% without any perceivable 
errour in the projection of Eclipses. 

8. To find the Moon's true horary motion from the Sun. 

With their respective anomalies take out their horary mo- 
tions from Table XVII. and subtract the Sun's horary motion 
from the Moon's ; the difference will be the Moon's true hora- 
ry motion from the Sun : in the present case 30* 49". 

, Now collect these elements together for use. 

D H M S 
1. True time of Full Moon in Aftril, 1819 10 1 1148 



2 r . Moon's horizontal parallax, 57 20 

3. Sun's semidiameter 16 4 

4. Moon's semidiameter, 15 38 

5. Semidiameter of Eearth's shadow at the Moon, 41 26 

6. Moon's true Latitude, North descending, 2 41 

7. Angle of the Moon's visible path with the > 5 #5 o 

Ecliptick, 5 

8. Her true horary motion from the Sun. 30 49 

These Element* being accurately prepared for the construc- 
tion of the Moon's Eclipse in April 1819, proceed as follows : 

Make a scale of any convenient length, A B, Fig. 3. and di- 
vide it into 60 equal parts, each part answering to a minute of 
a degree. « 

Draw the right line A B, (Fig. 4.) for part of the ecliptick* 
and R D perpendicular to A B for the northern part of its axis ; 
the Moon having North Latitude. 

Add, the semi diameters of the Moon and Earth's shadow to- 
gether, which, in this Eclipse, will make 56' 4" ; and take this 
from the scale in your compasses, and setting one foot in the 
point where the axis R D of the Ecliptick meets the right line 
A B as a centre, describe the circle AJD E N O ; in one point 
of which the Moon's centre will be at the beginning of the 
Eclipse, and in anpther point opposite to the former, at the end 
of the Eclipse. 

N. B. If the Moon's NorthL atitude had been equal to twice 
her semidiameter, it would have been sufficient to describe on- 
ly the semicircle ADEN. 

But in case her Latitude had been South, and equal to twice 
her semidiameter ; we must have described the semicircle N« 
O A. When her Latitude (whether North or South,) is less 



ARITHMETICS 201 

than twice her semidiameter, it will be best to describe a com- 
plete circle, as in the Plate, fig. 4. 

Take the semidiameter of the Earth's shadow, 41* 26'% in 
your compasses from the scale, and setting one foot in the same 
point for a centre as before, with the other describe the circle 
W L Y M for the whole circumference of the Earth's shadow 
at the Moon, through which she passes at her full, A fir it 1819. 

Make R D the radius of a line of chords on the sector, and 
set off the angle of the Moon's visible path with the Ecliptick* 
3° 35', from D to E ; (or, by dividing the quadrant DEN into 
90 equal parts, as in Fig. 1) and draw the right line T E for the 
northern half of the axis of the Moon's orbit, lying to the right 
hand from* the axis of the Ecliptick R D, because the Moon's lat. 
is North descending. It would have been the same way (on 
the south side of the ecliptick) if her Latitude had been South 
descending ; but contrary in both cases, that is, to the right 
hand from the axis of the Ecliptick ;■ if her Latitude had been 
either North attending or South ascending. 

Take the Moon's Latitude, ?* 41", from the scale, in your 
compasses, and set off that extent from the point in which the 
perpendicular R D falls upon the right line A W B,te T in the 
axis of the Moon's orbit ; and through the point T, at 
right angles toT E, draw k the right line P R T N for the path 
of the Moon's centre. 

Then, T shall be the point in the Earth's shadow, where the 
Moon's centre is at the middle of the Eclipse ; the midle point 
between R and T (which was not designated for want of room,) 
will be the point where her centre is at the Tabular time of her 
being full ; and R, the point where her centre is at the instant 
of ecliptical opposition. 

Take the Moon's horary motion from the Sun, 30 * 49 ", in 
your compasses from the scale A B (Fig 3.) and with that extent 
make the small scale (Fig. 5.) an divide it into 60 equal parts, 
or horary minutes* — Then as the true time of Full Moon in 
April 1819, was at 1 1 minutes 48 seconds, or 1 If minutes past I 
o'clock; take ll£ minutes from the (last mentioned) scale in 
your compasses, and set that extent from the point, signifying 
the instant of Full Moon (which is mid-way between R and T) 
to the left on the line (P R T N) of the Moon's centre, so shall 
that extent fix the point where the centre of the Moon is at the 
instant when it is I o'clock oX London. 

From this pointl, with the whole length of the scale (Fig. 5.) 
in your compasses, make marks along the whole length of the 
line in the path of the Moon's centre, and set the hour letters to 
these marks, as in the figure : then divide each space from mark 
-to mark, into 60 equal parts or horary minutes, as in (Fig. 5.) 



2P2 LOGARITHMIOK 

Take the Moon ( 8 semidiameter, 15 * 38 ", in your compasses, 
from the scale A B, and with that extent, as a radius, upon the 
points N> T, and P, as centres, describe the circle Q for the 
jVIoob at the beginning of the Eclipse, when she touches the 
Earth's shadow at Y ; the circle R for the Moon at the middle 
of the Eclipse, and the circle S for the Moon at the end of the 
Eclipse, just leaving the Eauth's shadow at W. 

The point N denotes the instant when the Eclipse begins, 
namely, at 25 minutes 30 seconds after XI in the morning : the 
point T the middle of -the Eclipse, at 10 minutes 18 seconds 
past I o'clock in the afternoon ; and the point P the end of the 
Eclipse, at 58 minutes after II. — Thus it appears, that the Moon 
was totally eclipsed for the space of 2 hours, 42 minutes, SO 
seconds. 

MOnJS EXAMPLES* 

Exp. Let it be required to find the Elements for the Solar 
Eclipse which happened in AjiriL 1764, New Style. 

D. H. M. S. 
I. True New Moon Afiril, 1764. 1 10 30 25 



2. Semidiameter of the Earth's disk, 

3. Sun's distance from nearest solstice, 

4. Sun's declination, North. 

5. Moon's Latitude, North ascending. 

6. Moon's horary motion from the Sun; 

7. Angle of the Moon's visible ? 

path with the Ecliptick. J 

8. Sun's semidiameter, 
S. Moon's semidiameter, 
# 10. Semidiameter of the Penumbra. 
This Eclipse was nearly central, and annular.* 



o 


9 


M 





54 


43 


77 


49 


53 


4 


49 








40 


18 


27 


54 


5 


35 







16 


6 




14 57 




31 


S 



* Note. In annular eclipses, the light of the Sun is left all 
around the Moon in a circulr form. Annular, from the Latin 
ctnnulusy a ring. 



9 


> 


*> 





57 


20 





IS 


56 





15 


59 





41 


34 





32 


21 


5 


35 


Q 





30 


52 



ARITHMETICS. TO 

"Exp. Let it be required to find the Elements for the Lunsp 
Eclipse in May, 1762, N. S. 

D, H. M, S. 
1. True Full Moon in May, 1 7 62. 3 5 50 50 



2. Moon's horizontal parallax, 

3. Sun's semidiameter. 

4. Moon's semidiameter. 

5. Semidiameter of Earth's shadow? 

at the Moon. y 

6. Moon's true Latitude, South descending. 

7. Angle of the Moon's visible > 

path with the Ecliptick. } 

8. Her horary motion from the Sun. 

Ex. 3. Required the Elements for the Eclipse of the Sun, 
^Ari/ 24th, 1819 ? 

jEx. 4. Required the Elements for the Lunar Eclipse, Oct. 
'3d, 1819? (total.) 

Ex. 4. Let it be required to calculate the Elements for the 
Lunar Eclipse, March 29th, 1820 ? 

Ex. 6. In the year 1823, there will be four Eclipses ; namely, 
two of the Sun, one February 11th ; and th* other July the 8th z: 
and two of the Moon, one January 26th, and the other July 23d* 
(both total.) Let it be required to tind the respective Ele* 
ments for the construction of these Eclipses ? 

Ex. 7. In the year 1826, there will be two Eclipses of the 
Moon ; viz. May 2 1st, and November 1 1th, (both total.) What 
are the Elements belonging to each I 

Ex. 8. What are the proper Elements for constructing apt 
Eclipse of the Sun, which will happen July 29th, J980, Okl 
Style ? . 

To find the number of Eclifises there are in any given year, 
•and in what Months they hafifien. 

Precept. Enter Table XVIII. and take out the mean Lon- 
gitude of the Moon's Nodes for the given year ; with which en- 
ter Table XIX. and find, in that table, when the Sun'-s Longw 
tude will be nearly the same of six signs different ; and the day 
of the month in which these numbers are so found, will be the 
time required. 

EXAMPLES. 

1. It is required to find the number of Eclipses in the year 
1796 ; and in whut months they will happen. 

The mean Longitude of the Moon's* Noi th Node, on the first? 
of January 1796, is S5 20° 35', of the South Node (it being just 



l 



204 LOGARITHMICK. 

six s^gns distant) 9S 20° 35* ; wherefore the Node-mtmtbs- are 
January, July , and December ; consequently there were three 
Eclipses in that year. 

2. Required the number of Eclipsed in the year 1 800, and in 
what months they happened. 

3. Required the number of Eclipses in the year 1820, and the 
months in which they happen. 

To find on what fiart of the Globe any given Eclipse of the 
Sun or Moon is visible. 

This is most readily ascertained by means of an artificial 
globe ; as follows : 

The day and hour being given when a Solar Eclipse will hap- 
pen y to find where it will be visible. 

Precept. Find the Sun's declination, and elevate the pole 
agreeably to that declination ; bring the place, at which the 
hour is given, to that part of the brass meridian which is num- 
bered from the equator towards the pole*, and set the index of 
the hour circle to twelve ; then if the given time be before 
noon, turn the globe westward till the index has passed over as 
many hours as the given time wants of noon ; if the time be past 
noon, turn the globe eastward as many hours as it is past noon, 
and exactly under the degree of the Sun's declination on the 
brass meridian you will find the place on the globe where the 
Sun will be vertically eclipsed : at all places within 70 degrees 
of this place, the eclipse may* be visible, especially if it be a 
total eclipse. 

Ex. On the 11th of February 1304, at 27 min. past 10 o'clock 
in the morning at London, there was an eclipse of the 
sun ; where was it visible, supposing the moon's penumbral 
shadow to extend northward 70 degrees from the place where 
the sun was vertically eclipsed ? 

Ans. Britain, Ireland, France, Germany, &c; 

The day and hour being given when a Lunar Eclipse will hap- 

pen tofihd where it will be visible. 

Precept, Find the Sun's, declination for the given day, and, 

note whether it be north or south ; if it be north, elevate the 

south pole so many degrees above the horizon a* are equal to 



•Note. When the Moon is exactly in the node, and when the 
atfis of the Moon's shadow and penumbra pass through the cen- 
tre of the earth, the breadth of the earth's surface under the pe- 
numbral shadow is 70° 20* ; but the breadth of this shadow is 
variable ; and if it be not accurately determined by calculation^ 
it is impossible to tell by the globe to what extent an eclipse of 
the sun will be visible. 



ARITHMETICS. 2*fr 

*lre declination ; if it be south, elevate the north pole in a simi- 
lar manner ; bring the place at which the hour is given to that 
part of the brass meridian which is numbered from the equator 
towards the poles, and set the index of the hour circle to 
twelve ; then, if the given time be before noon, turn the globe 
westward as many hours as it wants of noon ; if after noon, turn 
the globe eastward as many hours as it is past noon ; the place 
exactly under the degree of the Sun's declination will be th6 
antipodes of the place where the Moon is virtically eclipsed* 
Set the index of the hour circle again to twelve, and turn the 
globe on its axis till the index has passed over twelve hours ; 
then to all places above the horizon the eclipse will be visible; 
to those places along the western edge of the horizon the moon 
will rise eclipsed ; to those along the eastern edge she will set 
eclipsed ; and to that place immediately under the Sun's decli- 
nation the Moon will be virtically eclipsed. 

Example. On the 26th of January 1804, at 58 tnin. past seven 
in the afternoon, at London, there was an eclipse of the Moon ; 
where was it visible ? 

dnswer. It was visible to the whole of Europe, Africa, and 
the continent of A sia. 



27 



» 



\ 



A 

NEW AND CORRECT 

TABLE OF LOGARITHMS 

OF THE NATURAL NUMBERS FROM 1 TO 10.00#, 

EXTENDED TO SEVEN PLACES BESIDES THE INDEX ; AND S* 
CONTRIVED, THAT THE LOGARITHM MAY BE EASILY 
FOUND TO ANY NUMBER BETWEEN 1 AND 
10,000,000. 



ARITHMETICK 



2*7 



>* - 



1 U.JJJ'JJOO 

2O.3JU30u 
3047712i3 
40.602J60J 
5 0.6)8), oj 



o0.//6i5io 
70.3450980 

8 0.9030906 

9 0.954242o 
10 l.OOv/OOOO 



lM.im^;- 

12 1.0791812 
131 1139434 

14 1 1461230 

15 1 1760913 



16 1-204.1JJJ 

17 12304439 
Id 1.2552725 

19 1 2787536 

20 1.301030J 



No. 



Log 



N T O.I Loir i|X, 



Lop-. 



1.7075702 | 
1.7160033 
- 1.7242759 
54)1.7323938 
55; 1.7403627! 



jo 1 74«18tt0 | 
571.7558749' 
5hjl 7634280 ' 
59,1.7708520 l 
6017781513! 



01,1.7853298: 
62 1.8923917! 
631.7993405 ! 
64 1.8051800 I 
J35 1.8 12913 4 ' 

O6l.81954o3 
67U.826074S 
631.8325089 
69(1.8388491 
70 1.8450^)80 i 



J i 1 322^i.'J 

22 1.3424227 

23 1.3617278' 
241.3802112 

25 1.39/Q409 



< i 1.8512jo3 
72 ( 18573325 
73; 1.8633229 
74 1.8692317 
751.8750613 



2o 1.4i4'J; JJ 

2714313638 

28 1447*580 : 

29 14623980; 
301-4771213; 



31 149-ljol/. 

32 1.5051500 

33 1 5185139 

34 1 5314789 
.3) 1.54406H0 



(o l.edJ8l6(> 
77jl 8864907 
78' 1.8920946 
79,1.8976271 
80.1.9030900 



8i:1.90«4bi0 I 
921.9138139 
831 9190781 J 
84^19242793 , 
85 1.9294189 ! 



oj \.5563\Jzj\ 
37.1.5682017! 

38 1.5797835! 

39 1.5910646" 

40 1.60206.)0; 



80 1.9344J8J 
8719395193.' 
881.9444827'! 
89)19493900; 
901954? t^v 



+ • 1.61 j/ oj 
42 1 62321.93 
43:1.6.134685 
44.1.6434527 
45 1.6.5331 ?5 
*->, I 6.32,- ^c .) 
47! 1.5720979 
4816812412 
49.16901951 
5oil 696') 7 00 



yi 1.95r/j-i: •<• , 
921.963:378 j 
93 196848291 
941. 73127? 

95l9/7'2- 



97 1986771", 
981.9912261 
99 j 1995 53 52! I 
1U.r20j0iOK)| 



01 2.0043214 

02 2.0086002 
.03^2.0128372 
104;2.0170333 

05'2.0211893 



oo 2.0253059 
0720293838 

08 2 0334238 

09 2 0374265 

10 2 0413927 



11 2.045oJ50 

12 2.0492180 

13 2 0530724 

14 2 0569048 

15 2 0606978 



lo 2.0o44j6J 

17 2 0681859 

18 2 0718320 

19 2 0755470 

20 2 0791812 



zl 2 0827654 
22 2 0363598 
23,2.0899051 

24 2.0934217 

25 2 0969100 



26;2.1003/U5 
27,2.1038037 
28,21072100 
2912 1105897 
30,21139434 



31,3 1172/13 
32|2 1205739 
33;2123£516 
34:21271048 
35|2 1303338 



o6;2 1335389 
3721307206 
38| 21398791 
.3912.1430148 
40i21461280 



4l|il492i^i| 
42215228831 
43 I 21553360| 
44J215S3625 
45|2.16136rt0i 



46: 2.1 64352 w 
47 21 673173 
48121702617 

4i:| :.i73iS3o 

50|2 176;)913 



51:21789769 
52J21818436 
53J21846914 
5412.1875207 
55i21903317 



56:2 19312*0 
5721958997 
58 21986571 
59'2 2013971 
602.2041200 



Ln R . 



201 '2.3031961 

202 2.3053514 

203 2 3074960 

204 2.3096302 

205 2.3117539 



206 2.3138672 
2072.3159703 

208 2 3180633 

209 2 3201463 

210 2 3222193 



61 2.2w6o2oy 
6212 2095150 ' 
63 2 2121876 
6412 2148438 
65:2.217483' 



06:2.2201081 
67;2.2227165 
68;22253093 
69,22278867 
702.2304481- 



71 2.2329961 

72 2.2355284 
73.2 2380461 
742 2405492 
7512.2430380 



2ii2o242tt25 

212 2.3263359 

213 2,3283796 

214 2.3304138 

215 23324385 



2i6 2o344j38 

217 2 3364597 

218 2 3384565 

219 2.3404441 

220 2.3424227 



221 2 3443923 

222 2 3463530 

223 2.3483049 

224 2.3502480 

225 23521825 



762 245512, 
7712.2479733 . 
78 2 2504200 j! 
792 2528530 < 

80 2 2552725 [' 

81 22576786 

82 2.2600714 

83 2 2624511 

84 2.2648178 

85 2.2671717 



226 2.3541084 

227 2.3560259 

228 2.3579348 

229 2.3598355 
2 iO 2.3617 278 



86 2.2695129 
872.2018416 

88 2.2741578 

89 2.2764618 

90 2 278753 6 

yl 22810334 
92 2.28*301 
93,2 2855573 

94 2 287801 

95 229.00346 
yoj2 29226o7 
97!2.294466? 

98 2.2966652 

99 2 2988531 
2002.301030( 



2ol ,2 3636120 
232 2.36,54880 
233'2 3673559 
234J2 3692159 
23512.3710679 
"236-2 37 29120 
237J2 5747480 
2382.3765770 
2392.3783979 
24023802112 



241 2.3.820170 

242 2.38o8154 

243 2-335 06.: 

244 2 3873898 

245 2 38M661 



246 J.59u9o6l 

247 2.3926969 

248 2.394451 

249 2.3961993 
, 250 2 3979400 



I 



80S 



LOGARITHMICK 



"No, 

25l 

2522. 

253 



Log. 

2.3996737 

.4014005 

2.4031205 



259 

260 
261 



263 
264 



2542.404833; 
255 24065402 
25624082400 
2572.4099331 
25824116197 
2.4132998 
3.4149*33 
2.4166405 



2622 4183013 



2.4199557 
2-4216039 



265 ! 2.4232459 



366j2.4248816 



267 



2.4265113 



269 
270 



2682.4281348 



2.429752, 
2.4313638 



2/1:2.4329690, 
272 2.4345689! 
273|2.4361626 
27412.4377506 
275 ! 2.4393327 



27624400091 
277|2.4424798 
27824440448 
279.2.4456042 
280; 2.4471580 

2^4487063 
24502491 
2.4517864 
2842.4533183 
2852.4548449 



281 
282 
283 



286 
287 
288 



2.4563660 
2.4578819 
24593925 
289'24608978 
2902 4623980 



291 
292 
293 
294 
295 



2.4638930 
2.465382? 
2.4668676 
2.4683473 
2.4698220 



2962.4712917 
29724727564 

298 2 4742163' 

299 24756712 
3002.4771S13 



"301 
302 2. 



304 2. 



303 

304 

305 

306 

307 

3082. 

309 

310 



Log. 



2.4785665 
1.4800069 

2.4814426 
.4828736 

2.4842998 

2.4857214 
2.4871384 
.4885507 
2 4899585 
2.4913617 



311 

312 

31 

314 

315 



2.4927604' 
2.4941546 
24955443 
2.4969296 
24983106! 



316 
317 
318 



2.4996871| 
2.5010593 : 
„ 2 50242711 
3192.5037907 
320!2.5051500» 



32112.50650501 
322'2.5078559! 
323;2 5092025| 
32412.5105450 
325(3.5118834! 



32612.5132176 
3272.5145478; 

328 2.5158738! 

329 2.5171959; 
330:2.5185139! 



331 
332 
333 



2.5198280 
2.5211381 

25224442' 

33412.5237465 
33512.5250448 



336 



337 
338 



2.5263393! 



?. 5276299 
2.52891 o7 

339 2 5301997 

340 2.5314789 



341 
342 
343 
344 
345 



2.5327544 
2 5340261 
2.5352941 
2.5365584 
2.5378191 



346 
347 
348 
349 
350 



2.5390761 
2.5403295 
2.5415792 
2.5428254 
2.5440680 



No. Loy:. 



3512.5453071 
35225465427, 
353 2.5477747 
35412.5490033 
355 2 5502284 



356 
357 
358 
359 



2.5514500 
2.5526682 
2 5538830 
2.5550944 



3602.5563025 



361 
362 
363 
364 
365 



366 2 



367 
368 
369 
370 



371 

372 

373 

37412. 

375 



2.5575072 
2.5587086 
2.5599066 
2.5611014 

2.5622929 



5634811 
2.5646661 
2.5658478 ( 
2.5670264 
2.5682017 



2.5693739 
2.5705429 
2.5717088 
572871^, 
2.5740313 



3762.5751878 
3772.5763414 



378 
379 
380 



381 

382 

383 

3842 

385 



386 
387 
388 
389 
390 



391 
392 
393 
394 
395 



39/ 
398 
399 
400 



2.5774918 
2 5786392 
2 5797436 



2.5809250! 
2.5820634! 
2.5831988 
5843312 
2.5854607 



2.5865873 
2 5877110 
25888317 
2.5899496 
2.5910646 



2.5921768: 
2.5932861, 
2.5943926 
25954962 
2.5965971 



396(2 5976952 
2 5987905 
2 5998831 
2.6009729 
2.6020600, 



No. | Log. 



401 2.6031444 
4022.6042261 
4032.6053050 
4042.6063814 
405J2 6074550 



4062.6085260 
407J2 6095944 
4082,6106602 
409J2.6117233 
4102.6127839 



4112.6138418 
41212.6148972 
413 2 6159501 
4142.6170003 
415'2.6180481 



4162.6190933 
417|2.6201361 
4182.6211763 
4192.6222140 
4202.6232493 



421 2.6242821 



422 
423 
424 
425 



426 
427 
428 
429 
430 



431 
432 
433 
434 
435 



4362 

437 

438 

439 

440 

441 
442 
443 
444 
445 



446 



447 2. 



449 
450 



2.6253125 
2.6263404 
2.6273659 
2.6283889 



2 6294096 
2.6304279 
2.6314438 
2.6324573 
26334685 



2.6344773 
2.6354837 
2.6364879 
2.6374897 
26384893 
;! 



No.f Log. 



6394865 1 
2.6404814; 
2.6414741i 
2 6424645' 
26434527! 



26444386 
2 6454223 
26464037i 
2.6* 73830! 
2 6483600 



4512.6541765 
4522.6551384 
45326560982 
454 2.6570559. 
455*2.6580114 



4562.6589648- 
457J2 6599162 
458 2.6608655 
459.2.6618127 
4602.6627578 



4612.6637UU9 
4622.6646420 
46312.6655810 
464J26665180 
465 ! 2.6674530 



.'66|2.666ob59 
46712.6693169 
468*2 6702459 
469J2.6711728 
47012.6720979 



471 
472 
473 
474 
475 



2.6730209 

26739420. 

2.6748611 

2.6757783 

26766936 



476 
477 
478 
479 
480 



26776070 
2.6785184 
26794279 
2.6803355 
2.6812412 



481 
482 
483 
484 
485 



487 
488 
489 
490 



2.6821451 
2 6830470 
2.6839471 
2.6848454 
2 6857417 



486 2.6866303 
26875290 
2.6884198 
2.6893089 
2.6901961 



491,26910815 
4922.6919651 
4932.6928469 
4942.6937269 
495,2.6946052 



2.6493349 

.6503073 

448 2.6512780 

2.6522463 

2.6532125! 



496 2-6954817 



497 
498 
499 
500 



2.6963564 
2 697229: 
2.6981005 
2 69897Q0 



ARITHMETICS 



409 



501 

502 
503 
504 
JJ05 
50b 
507 
508 
509 
510 
611 
512 
513 
514 
515 
"51b 
517 



Lotr. 



2.699837! 

2.700703 

2 7015680| 

2.7024305 

270329H| 

2.7041505! 

2.7050080 

27058637 

2.7067178! 

2 7075702 



27084209 
2.7092700 
2 7101174 
2.7109631 
2.7118072; 
2.7126497! 
2.7134905 
51827143298^ 
519J2.7151674 1 
^201271600331 
521J27168377I 
5222.7176705 
52312.71850171 
524i27193313 
525; 2.72015 93- 
526|27209857i 
5272.7218106 
5282.7226339 
529;2 7234557S 
5302.7242759 



No. Log. No. Log. 
"55T2.74ll5l6f"B0l2! 

552 2.7419391 602 2. 

553 27427251 603 2 
55427435098 604 

555 27442930 605 

556 27450748, b06 

557 27458552' 607 
5582 7466342! 608 *■ 

559 2 7474118! 609 2 

560 2.7481880; 610 2 - 

561 2.7489629' 6li- 2 

562 27497363J, 619 2 

563 27505084|! 613 2 

564 27512791!| 614 2, 

565 27520484 ' 615 2. 

566 2 7528164;i~bl^2 

567 2 7535831' 617 2 
56827543483 618 2 



53127550945; 
532'2 7259116 
53327267272 
534 27275413 
53527283538 
~536 272yi64d 
53727299743' 
538 27307823 
53927315888 
540 27323938 
541 27331973 

542 27339993' 

543 27347998 1 

544 27355989i 
^45 27363965' 

546 2.7371926! 
547.27379873 
543 27387806 
549,27395723 
55027403627 



7788745 
!7795965 
7803173 
7810369 
7817554 
7824726 
7831887 
7839036 
7846173 



No 



651 
652 

I 65, 
654 
655 
006 
65," 
658 
650 



;'2.81358: 



,|No. Lo<-\ 



8135810 701 



2.8142476 
2.8149132 
2.8155777 
2.8162413 
2 8169038 
.'.8175654 
2.b782259 
2.8188854 



569 27551123 

570 27558749 , 
571 275663bl 
57227573960. 

573 27581546 

574 27589119 

575 2.7596678 

576 27604225, 

577 27611758! 
57827619278 

579 27626786 

580 27634280 



619 2, 

620 2 
T5l£ 

622 2 

623 2. 

624 2. 

625 2. 



631(2 

632 2. 

633 2. 

634 2, 

635 2 



~S36 2~ 

637 2.i 

638 2 

639 2. 



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3.1519824l!l469 



3152288l||1470 3 1673173 



3.152594lil471 
14223.1528996J 

3.1532049 
14243.1535|00 

3.1538149 



31541195^ 

.1544240 

142831547282 

3.1550322 
143031553360 



3.1556396 



14323.1559430 
143331562462 
1434:3.1565492 



1435 , 3.1568519 



1436,31571544 
143713 1574568 
143831577589 
143931580608 
144031583625, 



1441 3.1586640s 
1442 ! 31589653 

1443 31592663 

1444 31595672! 

1445 3.1598678; 



134631290451 13963.1448854 
1347(3.1293676 1397,3.1451964 
1348'31296899 1398 3.1455072 
13493. 13001 19 11399 3.145H177 
13503 1303338 11400 3.1461280 



31676127 
147231679078 
1473 31682027 
14743.1684975 
147531687920 



1383 
1484 
1485 



31670218 



147631690864 
1477 31693805 
147831696744 
1479 31699682 
148031702617 



1481 3.170555X 

1482 3170848S 
3171141: 
3.171433^ 
3.1717265 



1486 

1487 

14883, 

14893. 

1490 



1491 

1492 

1493 

14943. 

14953. 



172018* 
172311(3 
,1726029 
1728947 
1731863 



1734776 
1737685 
1740598 
1743506 
1746412 



1446[3.1601683| 14963.1749316 
144731604685 1497J3 1752218 
1448316076861 149831755118 
1449 3.1610684 14993 1758016} 
14503 1613680'|1500l3.1760913f 



ARITHMETIC*, 



SIS 



Nfo.| Log^ ! No. 

I501|3.176j«u/ i 1551 

1502 3.1766699 1552 

1503 3.1769590 1553 
1504&1772478J1554 
1505 3.17753651(1 555 



3.1804126 
15163.1806992 
151731809856 
1518 

15193.1815578 
15203.1818436 



1506 3.1// 8i>oU 

15073.178113S 

15083.17W01 

15093.1786892 

1510*3.1789769 



15113.1792645 



1512 
1513 



3.1795518 
3 1798389 



OTi 

1522 
1523 



15143.1801259 
1515 



3.18212*2 
3.1824147 
3.1826999 



11556 
!l557 
1558 
1559 
! 1560 



1561 
1562 
1563 



„og. 



3.1906118 

3.1908917 ( 

3.1911715 

.1914510 

1917304 

3.1920096 

1922fc86 

31925675 

3.1928461* 

3.1931246 



3.1934U29 
31936810 
3.1939590 



15643.1942367 
3.1945143 



1565 

iswi'o.iiMfmtt 

1567|3.1950690 

3.1812718| 1568 3.1953461 

15693.1956229 

15703.1958997' 



1606 

1607 

1608 

1309 

1610 

1611 

1612 

161 

1614 

1615 

1616 

1617 

1618 

1619 

1620 



57113.1961762, 
'1572 ! 3.1964525 
1573|3.1967287| 
1574 , 3.1970047 
1575J3.1972806 



1625 



32108534 



157613.1975562 
157713.1978317; 
15783.1981070 
15793.1983821 
1580 3.1986571 



lb* 
1627 



3.2119211 
3.2121876' 



15813.1989319, 
1582 3.1992065! 
!1583!3.1994809' 
158431997552, 
15JB5| 3.200029 3 
15863.2003032 
1587;3.2005769 
153835008505 
!l589!3.2011239j 
15903.2013971 1 



3.2111205 
12113876 
1628|3.2H6544 
1629 
1630 
1631 
1632 
1633 
1634 
1635 



3.2124540 
3,2127202 
3.2129862 
3.2132521 
3 2135178 
I5f6 3513783; 



1 15243.182985Q 
15253.1832698 
1526:3.1835545 
152731838390 
15283.1841234 
15293.1844075 
1530 31846914 

S53I 3.1aW:K 
15323.1852588 
1533'3.1855422 
153413.1858254 
1535 3.1861084 
I53S ! 3.1863912 
15373.1866739 
15383.1869563 
393.1872386 
15403.1875207 

1541 3.1878026 159l|3 2016702| T641 
15423.1880844 15923.2019431' 1642 

1543 3.1883659 11593 3.20221581 j 1643 

1544 3.1886473l!l594 3.2024883;! 1644 
1545 : 3.1889285 iil5953 : 2027607lll645 
1546 3.1892095,110^6 3.2030329,1046 >5l642!*8 j 
1547 3.1894903 1 1597,32033049: 1647 3 2166936 I 
1548 3 1897710 1598 3.2035768 1 ; 1648 3.2169572|| 
1549 3 1900514 159935038485', 1649 3 5172207|' 
15 50'319Q3317'll6QO 3.204 1200 1650 3.2174839, 

28 



1605 



Lop. 



No. 

lS6I 3.2043913! 

16023.2046625 

1603 3.2049335 

16043.2052044 1 



! 2054750' 



o.^Uj/4d5, 
3.2060159! 

2062860' 
3.2065560 

2068259, 



o.2U/Ui*jj 
3.2073650 
3.2076344 
3.2079035 
3.2081725, 
.2084414 
3 2087100 
3 2089785 
3.2092468 
3.2095150 



1621 3.2097830 

1622 3.2100508 

1623 35103185 

1624 3.2105860 



1637 
1638 
1639 
1640 



3.2053044 
3.2062860 
3.2065560 
3 2068259! 



o.2 la 1086 
35153732' 
35156376; 
32159018 
35161659 1 



No./ Log. ijNo, 
rW135T7747ii 
6523.2180100 
3.2182729 



653 
6543.2185355 



655 



32187980 



66; 

664 

665 

666 

667 

668 

669 

670 



1701 
1702 
1703 



3.230704^ 

3 2309596 

1.2514146 



1705 

3563.21906031 l'M> 

657 

658 

659 

£60 

661 



17043.2314606 



3.2317244 



S^owyo 



3.2193225 Il707 3.2322335 
3 2195845 J1708 3.2324879 
3.2198464 1709 

35201081 171? 
3.220369611711 



6623.2206310 !171i 



35208922 ;1713 
35211533 11714 
35214142,1715 



3.2216750! 1716 
3 2219356*1717 



35221960 
3.2224563 
35227165 



oil 
672 
.673 



35229764 

223236; 

35234959 



675 



67435237555 1 



35240148 



o/o«o 2242740 



677 
678 



35245331 
3.2247920 



67935250507 



35253093 



680 

681 35255677 

68235258260 

35260841 
2263421 

35265999 



689 
690 
b'jl 
692 
69; 
694 
695 



696 
697 



699 



1731 
1732 
1733 
1734 
1735 
3.2268576l?ob 



1718 
1719 
1720 



1721 



683 

684 

£85 

686 

68735271151ill737 

68835273724 1 1Z38 
3.227629611739 
35278P67iil740 



Lo?. 



35327421 
35329961 
3.23o2506 
35335038 
35337574 
3.2340108 
3534G641 
2345173 
35347703 
3 2350232 
35352759 
3 2355284 



S2357iiu* 



1722353603^1' 
172335362853 
172435365373 
1725 3 2366891 
1726o.2370408 
(1727 
1728 
1729 
1730 



35372923 , 

35375437 

2377950 

2380461 



3.23142971 
3.23885*47 , 
3.2387986 
35390491 
35392995 



352«14;>t5|il741 
3 22840Q4, 174: 
35286570 
3 2289134! 



352916971 174.* 



1743 

1744 



3.2oyo4V/ 
35397998 
32400498 
35402996 
35405492 

35410482 
35412974 
35415465 
32417954 



3.22*42.^111746 32420442 



35296818j|1747 
698352993771 174£ 



35301934)1749 



700l35304489i;i750 



3 2422929 

13.2425414 

{.242789* 

3543038J 



-*u 



IiOGARITHMrGK 



7513.2432861 
752(3.2435341 
1753{3 2437819 

I7543.244029f 
175513.2442771 



175613.2445245 
175713 2447718 
1758:3 2450189 
1759J3.2452658 
17603 2455127 



1762 
1763 
1764 



No. Lotf. 



1761 !3 2457594 



32160059 
2462523 

3 2454936 : 



1765 3 2467447 
l/oo t 3.2459907 t 
1767(3 2472365' 
176813.24/4823!' 
1769 3.3477278 
17703.2479733 
I77'l's.2482*36 
1772 l 3.2484^r 
1773^3.2487087 
17743.2489536 
1775 3 249J984 
17763.2494430 
3.2496874 
3.2499318 
3 2501759 
1.2504200 
3.2506639 
3.2509077 
3.2511513 
3.2513949 
1.2516982 



1777 
1778 
1779 
1780 

1781 
1782 
1783 
1784 
1785 



1786 
1787 
1788 
1789 
1790: 



3.2518815 
3.2521246 
3.2523.V5 
3.2526103 
3.2528530 



1791 
1792 

1793 
1794 
1795 
1796 
1797 
1793 
1799 
1800 



3.2530956 
.2533383 
3.2535803 
3.2538224 
3 2540645 
3.2.543 octf 
3.2545481 
3.2.547.397 
3.2530312 
3.2552725 



No 

~801 3 

.8023 
803|3. 
804J3 
805|3 
80613" 
8073 
8083 
8093 
810!3 



Loe. 



2555137 
255754F 
2559957 
2562365 

2561772 
"2567177 
2569585 
2571984 
2574386 
257678C 



595930(1868 




.3603uy9 
.2605484 
.2607867 
.2610248 
.2612629 
261j(JUd 
.2617385 
2619762 
,2622137 



No. | L.O£. 

1861 3" 
18523 

1853 3. 

1854 3 

1855 3 



2674064 

2576410 
2678754 
2681097) 
2683439 



ibooo 

1857 3. 

1858 3. 

1859 3 
18603 
I3o73. 
1362 3 
1863 3. 
1364 : 3. 
1865;3 
1866'3 
1867|3. 

3 
1369' 



1870 j3 

11871J& 
H872'3. 
1873 3. 
1874,3. 
18753. 



.26857 8U| 
,2688119 
.2690457 
2692794 
26? "51 29 
.2697464' \ 
2699797 
2702129 
2704459 ; 
27 06788 
2709116 
2711443 
2713769 
2716093 
271841 6( 



901 
902 
90^ 
904 
905- 



8323. 
8333. 
834 j 3 
835 ! 3 

836 3 



la/63. 
1877J3- 
18783- 

~™^. 1879,3. 

2624511 1880.3. 
18813: 
18823. 
1883'3. 
18843 
1885'3, 



2626883 
2629255 
2631625 
2633993 
2636361 



83713. 
8333. 

3393 
840J3. 

34113" 
842;3. 
8433 
?>14\3. 
845 3. 

846 
847 
848 
849 
850 



26387271 ; 

-2641092' 

.2643455; 

2645817 

2648178 

2650538 

2652896; 

2655253 

2657609 

2659964 

1.2662317 

I.2664C99 

1.2667020 

i.2669369 

1.2571717 



272u/o8» 
2723058 
2725378 
2727696 
273001 



.2*32328 
2734643 
2736956 
.2739268, 
2741578 



27438881 
2746196! 
2745809- 
2750809 
,2753114! 



18863. 
1887)3. 

18883. 
1889:3. 
1890J3 
1891J3"! 
18P2'3. 
1893J3 
18943. 
1895J3. 
1896.3. 
18973- 
1898'3 
1899(3. 
1900 ! 3. 



.2755417 
,2757719, 
2760020 
.2762320 
27646181 



.2766915 
.27692111 

2771506 
.2773800; 
2776092; 
2778383 
2780573,' 
2782962- 
2785250; 
27875361 



32789821 
3 2792105 
3 2794388 
3 2796669 
3 2798950 



9063.2801229 
907|3 2803507 
9083.2805784 
909'3.2808059 
910 3 281033 4 
:^ll!3"2812607. 
912 3.2814879' 
9133.2817150 
914!3.2819419 
915J 3.282168 8 
yl 6^3.2823955 
917J3.2826221 
918:3.2828486 
9193.2830750 
920J32833012 
92132835274 
922;33837534 
923(32839793 

924 3.2842051 

925 3.2844307 



1926 
927 



3.2846563 
3.2848817 



19283.2851070 



3.2853322 
3 2855573 



1932 
933 
934 
.935 
L936 
1937 
938 



929 
L930 

9313 2857823 



3.2860071 
3.2862319 
3.2864565 
32866810 
3.2869054 
3 2871296 
3.2873538 
.939(3.2875778 
.9403.2878017J 
.94113.2880255 
.942*3.2832492 
.943'3.2884728 
.944 3.2886963 
l945' 3.2889196 . 
[9463.28914291 
9473.2893660} 
.943'3.2895390 
1949 ! 3.2898118 
.9503.2900346' 



No.| Log. 

19513.290257 

195232904798 

1953 3.2907022 

19543.2909240 

19_5_5'3.2911468> 

1956I3.2913689 

1957J3 2915908 

1958 3.2918127 

1959 3.2920344 

1960 3 2922561 



1961 
1962 
1963 



4966 
1967 
1968 
1969 
1970 



3.2924776 
3 2916990 
3.2929203 
1964,3.2931415 
19653 2933626 
3.2935835 
3.2938044 
3.2940251 
3.2942457 
3.2944662 



1971 3.2946866 
1.972J3 2949069 
1973 3.2951271 
1974 ! 3 2953471 
197513 2955671 



19763.2957869 



1977 
1978 



3.2960067 
3.2962263 



19793 2964458 
1980 3 2966652 
laol',3 2968845 
1982'3.2971037 
1983 3.2973227 
1984'3.2975417* 
1985 3.2977605 



1986 
1987 
1988 



1991 
1992 
199: 



0.2*79792 
3.2981979 
3.2984164 
19893 2986348 
19903.2988531 



3.24/90713 
3.299289: 
3.2995073 
1994,3.2997252 
1995 3 2999429 



199b 
1997 
1998 
1999 
2000 



O.5UU1605 
33003781 
3 3005955 
3.30081 2b 
3.301030C 



ARITHMETICK 



«15» 



>»., 



Log. 



2001 3.3012471 
20023.3014641 
2003 3.3016809 
20043.3018977 
20053.3021144 



2006 3.3023309 
20073.3025474 
20083.3027637 
2009 3.3029799 
20103 3031961 



20113.3034121 

2012 3 3036280 

2013 3 3038438 
20143 3040595 
2015 3 3042751 



2016 3.3044905 
10173.304705^ 
20183.3049212 
20193.3051363 
2020 3.3053514 



2021 3.3055663 
20*3.3057812 
2023 3 3059959 
20243.3062105 
202533064250 



2026 33066394 

2027 3.3068 37 

2028 3.3070680 
20293.3072820 
20303 3074960 



No. 



2051 
2052 



LOVT. 



3.290257. 
2904798 



2053 3 2907022 



21U1 
2102 
2103 
2104 
2105 
2056(3 2913689 |2106 



2054 



3.2909240 



2055,2.2911468 



.2057 
'2058 
2059 



No. | Log-. 



3.3224261 
122632? 
3.3228393 
3.3230457 
3 3232521 



3.2915908 2107 

2918127,12108 

3.2920344 2109 



206032922561 3U0 



3.323438-1 
3 3236645 
3.3238706 
3.3240766 



20613.2924776 

2062 3.2926990 

2063 3.2929203 
4, 

2065|3 2933626 121 15 



3.3244682 
3.3246939 
3 3248995 
206413.2931415 |2114 3 3251050 
33253104 



,2111 
J2112 
21JS 



200613,2^35838 
2067 13.293 8044 
'2068 3.2940251 
2069>3.2942457 
20703 2944662 



|2071 
2072 
207: 



3.2946866 
3.2949069 
3.2951271 
2074 3.2953471 



(2076 

2077 

2078 

|2079o 

2080*3 



2121 

2122 
2123 
2124 
2125 



3.2957869 
3.2960067 

.2962263 
3.2964458 

2966652 



20313.S077099 
2032 3.3079237 
20333.3081374 
2034'3.3083509j i 2084|3, 
2035'3.3085644! l 2085i3 



2036 3.3087778! <2086 ! 3 
20373 3089910:20873 
20383.3092042 2088 3, 
2039:33094172 20893. 
2040 3.3096302|!2090;3. 



,208113 3182721 

[20823.3184807 

2083|3.3I8S893 

.3188977 

.3191061. 



2116 
117 

2118 
119 

2120 



2151 
2152 
2153 



3 3242825 31 60*3 334453 8 
2161 :j 3346548 
2162:3 3348557 

2163 3.3350565 

2164 3.3352573 

2165 33354579 



3.325515/ 
33257269 
3.3259260 
3 3261310 
3263359 



3 3265407 
3 3267454 
3.3269500 
3.3271545 
3273589 



2126 

2127 
2128 
2129 
21 



33275633 

3.3277675 

3279716 

3 3281757 

83796 



3193143 
.3195224 
.3197305 
.3199384 ]2139;3 
3201463 2140 3 



21313 

21323 
21333 
21343 
2135J3 
2136'3 
21373. 
2138'3 



2041j3.2880255 
2042'3 2882492 
2043|3.2884728 
2044'3.2886963 
2045!3.2889196 



2046, 
2047 
£048 



?049 3 



3.2891429 

3.2893660 

2895890 

2898118 



,2050 



2091 
2092 
2093, 
2094 
i2095 



i2096 
'2097 
J2098 
12099 



3 2900346 ) I2100 



1.3203540 
1.3205617 
1.3207692 
1.3209767, 
i.3211840 



3.3213913 
33215984 
33218055 
3.3220124, 
3.3222193 1 



3285834 
3287872 
3289909 
3291944 
3293979 
i3296012 
3298045 
3300077 
S302108 
3304138 



21413.3306167 
3142&3308195 
2143*3 3310222 
2144'3.3312248 
214513.3314273 



21467.3316297 
214713.3318320 
21483.3320343 
2149,33322364 
2150'3,332438. 



1-ujC. 



3.3326404 
3 332842, 
3.3330440 

2154)3.3332457 

[1553.333447, 

2156 

J157 



3336484 
3 3338501 
2158 3.3340514 
2159|3 3342526 



2106 3 3356585 

2167 

2168 

2169 3.3362596 

2170 3 3364597 



3 3358589 
3 336059. 



2171 3.3366598 

2172 3 3368598 
2173J3.337059/ 
12174*3.3372595 
!2175 ! 3.3374593 



2176,3 o3/ 6589 
!2177i3.3378584 
2178 , 3.3380579T 
217913 3382572 
21'80 ! 3 3384565 



2181 ! 33386557 
2182 ! 3 3388547 
2183J3.3390537 
21843.5392526 
21853.3394514 



21863, 
21873 
2188 3 
2189:3. 
|21903. 



.3396502 
3398488 
3400473 
S402458 
3404441 



21913 
21923 
21933 
2194'3, 
219513 



34U6424 
3408405 
,3-410386 
.3412366 
.3414845 



2196 3.3416323 

2197 3-3418301 
3198.3.3420277 
2199 3 3422252 
220O3 342422; 



No. I Log. 



J2201I3.3426200 
i2202'3.3428173 
2203 ! 3^430145 
22043 3432116 
2205 ! 3 3434086 



2206 0.343 6055 

2207 3.S438023 
2208 ! 3.3439991 
22093 3441957 
?210'3.S443923 



2211 o.o445887 
2212'3.3447851 

2213 3.34*vS14 

2214 3.3451776 

2215 3 3453737 



2216oo455698 
2217 3 3457657 
22183.3459615 
2219! 3.3461573 
2220' 3 3463530 



22213.3465486 

2222 3.3467441 

2223 3.5469395 

2224 3.3471348 
222533473300 



222633475252 

2227 3.3477202 

2228 3.3479152 

2229 3.3481191 

2230 3.3483049 



2231 
2232 
2233 

2234 
2235 



3 5484996 
3 3486942 
3 3488887 
3.3490832 
3492775 



2236.o 
223713 
22S6 , 3 
2239*3. 

22403. 



o494718 
3496660 
3498601 
,3500541 
,3502430 



.35U4419 

3506356 

35082! 

35102! 

3512163 



2241 13. 
22423. 
■2243*3. 
2244.3 
2245.3 

2246 3 3514098 

2247 5 3516031 

2248 3.3517963 

2249 23519895 

2250 3 3521825 



216 



LOGARITHMICK 



2251 

2252 

22533 

22543 

2255 



NZ Log- |XO-l log- li No -' L{ ^1 



225613 

2257 

22583 

22593 

2260;3 



3523753 
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26013,4151404 

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2609 

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2611 



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34092567 
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26513.4234097 
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27 16 3 .4339398 



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3.4344092 
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27213 



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268113.4282968 

2682 3.4284588 

2683 3.4286207 
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3 4385423 



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3.4390167 
3.4391747 
13 4393327 



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4504031 
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2876 
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2783 3.4^45132!!2833;3 4522966 

2784 3 4446692;!2834'3 4523998 
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2831 
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3.4531654 2889 



3 4533183 



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2796344653^2 2846 3.4542349 
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27983.4468477i 28483.4545400 
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58003.4471 580, 12850 3.454S449 1 . 



2886 
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2890 



2891 
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3.4553018 
3.4554540 
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No.f Lo£. |,No| 



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3.4560622 
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2901 3.46254/71,2951 3.4699692 

2902 34626974 1 2952 O.4701164 

2903 S.46^8470 1 2453 3.4702634 
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2906 , 3.463295b,-2956 3.4707044 

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2916 3.4647875 29663 4 
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2871 3.4580332 

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2874 3.4584868 

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3 4587889 ;2926 3 4662743 
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2976 
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3.4593925|!2930 3.4668676! 2980 



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3011 
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3016 
3017 
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3020 



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3002 
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3 4787108 
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30563.4851533 
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3060'3.4857214 



31013.4915018 
1023.4916418 
3103 3 4917818 
31043.4919217 
31053 4920616 



3061 
3062 
3063 



3.4858633 
3.4860052 
3.4861470 
130643.4862888 
1396534864305 



|3066 
067 
068 
069 

3070 



3071 
3072 
1073 
3074 
3075: 



30363. 
3037|3. 
3038" 
3039 
1040 



3.4865722 
3.4867138 
3.4868554 
3.4869969 
3.4871384 



3.4872798 
3.4874212 
4875626 
3.4877039, 
3.4878451' 



3076 
077 
3078 
3479 
3080 



3081 
3082 
308: 
1084 
1085 



4823018 30863. 
4824448J 3087 
.482587813088 
4827307*3089.,. 
4828736T30903. 



3041 

3042 

3043|3. 

^044-3. 

3Q45J3 

3046J3" 
304?3. 
3048j3 
30493, 
>3050'3 



3 4830164! 3091 
3 4831592 



31063.49^2015 
31073.4923413 
31083.4924810 
31093.4926207 
31103.4927604 



31113.4.929000 
31123.4930396 
3113 3.4931791; 
31143.4933186 
3115 3.4934581 



3116 
117: 

3118 
119 

3120 



.4879863' 
.4881275 
.4882686 
.48840971 
.4885507 



.4886917) 
4888326 
4889735 1 
4891144 
4892552' 



I.4893959J 
1.4895366, 
1.4896773-! 
1.4898179! 
1.4899585' 



4833020 
4834446 
4835873 



.4837299 
.4838725 
4840150 
.4841574 
4842998 



3092 
309. 
3094 
3095 



3096 
3097 
3098 
3099 
3100 



1.4900990 : 
1.4902395! 
14903799; 
1.4905203: 
149066071 



.4908010 l 
4909412. 
.4910814' 
4912216^ 
,4913617" 



No., Loe. 



3151|3.49iJ4484 
3152i 3.4985862 
31.53 ! 3.498724 
3154'3498861 
31553 4989994 



3156 3 4991370 
3157*.°4992746 

3158 3.4994121 

3159 3.4995496 

3160 3 4996871 



3.4935974 
.4937368 
3.4938761 
3.4940154 
3.4941546 



312113.4942938 
31223.4944329 
31233.4945720 
31243,4947110 
312534948500 



126|3, 

1273. 

31283, 

31293. 

130J3. 



,4949890 
4951279 
4952667 
.4954056 
,4955443 



Noj 



l^o<. 



31613 499*245 
1623.4999696 
163 3 5000992 
1643 5002365 1 

3165 3 500373; 



No. Log 1 . 



li^Ol 
3202' 
3203,' 
3204 
3205 



O.D052857 
3.5054218 
3.5055569 
3.5056925 
3 505828c 



3.5059635 
3 5060990 
3 5062344 
3.5063697 
3.5065051 



3166 3.5UU5109 
31673 5006481 

3168 3.5007852 

3169 3.5009222 

3170 3.5010593 



31713.5011962 
3172'35013332 
'3173 3.5014701 
J3174 3.5016069 
3175 3.5017437 



131)3. 
3132*3. 
31331.° 

3134!: 



1135I3 



,4956831 

4958218 
.4959604 
.4960990 
4962375 



3136|3. 
;i373, 
11383. 
31393 
*140.3 

14ll4. 

142(3. 
3143j3 
3U4J3. 
5j3, 



4963767 
.4965145 
,4966529 
4967913 
4969296 



4970679 
4972062 
4973444 

4974825 
,4976206 



3146! 

1473 

31483. 

9[3 

3150i3 



4977587 
4973Q67 
4980347 
314913 4981727 
4983106' 



3176 

177 

178 

3179 

3180 



3181 
3182 
183 
3184 
3185 



|3186 
3187 
4188 
3189 
3190 



3.5018805 
3.5020172 
3.5021539 
35022905 
3.5024271 



3,502563; 
3.5027O02 
3.5028366 
3.5029731 
3.5031094 



8206 
3207 
3208 
3209. 
3210 1 

321l';3.oo6fc4l>i. 
321213 5067755 
3213;3.506910? - 
3214 3.507045S 
8jl5; 3.5071810 fc 

16 3.507316C 



3217 
3218 
3219 
3220 



3.9074511 
3,5075860 
3 507721C 
3 507855S 



3221 
3222 
3223 
3224 
3225 



3 5079907 
3 5081255 
3508260J 
3.5083950 
3.508529; 



3226 
3227 



3229 
3230 



3231 



3233 
3234 
3235 



3.5032458 
3.5033821 
3.5035183 
3.5Q36545 
3 5037907 



^237 
'323S 
[3239 



3.5086644 

3.5087990 

3228 3.5089335 

3.5090680;. 

3.5092025 



3.509337C 



323235094714 



35096057 
3.509740C 
3509874S 



32364.5100085 



3 5101427 
&5102768 
3.5104109 



|o2^0 3.5105450 



.3191 a 

3192 S. 

3193 3. 
131^3, 
13195J3. 



5U3926S 
5040629 
5041989 
5043349 
5044709 



3196-3. 

3197)3, 

3198 3 

3199 3 
S?0f>\ 



5046068 
,5047426! 
5043785 
505C142 



324l|^51«v90 
3242J8.5I08130 

3243 3.5105469 
8244 35110608 
3245 3.5US14? 



o246(b 51 13485 
3247 3.51 14823 
3248)3.5116160' 
3249[3 51174PT 
3?50"3.5!1J!834? 



1 



220 



LOGARITHMIC* 



No. 



Log. 



3551 3.5120170 
325255121505 
3253 3 5122841 1 
3254 ! 3.5124175' 

3255 3.512551Q 

3256 3.5126844 

3257 3.5128178 

3258 3-5129512 
325£ 3.5130844' 
3260 3.5132176' 



No.| Lop. 



33 jl 3.5186455 
33023^187771 
33033.5189086 
33043.5190400 
3305 3 5191715 



33063.519302S 
33073.5194342 
33083 9195655 
33093.5196968 
33103.5198280 



3261 & 5133508 
32623513-1840' 

3263 3.51361711 

3264 3 5137502; 

3265 3.513883 21 

3266 3.5140162J 
3267,3.5141491 

3268 3.5142820 

3269 3.5144149 



3311 3 5199592 
33123.5200903 
3313 3.5202214 
33143.5203525 
33153.5204835 



33163.5206145 

3317 35207455 

3318 3.5208764 

.3319 3.5210073 

3270 3.5145478! 3320 3.521 1381 



3271 3 5146805 
3272 35148133 
32733.5149460 
32743.5150787 
3275 35152113 



32763.5153439' 
3277 3.5154764* 



8278 
3279 



3.5156089 
3.5157414 



32803.5158738 



C28l|3.5160062 

5161386 

3.5162709. 

3.5164031J 

3.5165354 



3282 3. 
3283 
3284 
3285 



3286 
3287 
3283 
3289 
3290 



3291 
3292 
3293 
3294 
3295 



3296 
3297 
3298 



3326*3.5219222 
33273.5220528 
33283.5221833 
33293.5223138 



3330)85224442 



33313.5225786 



3.5166676 
3.5167997 
3.5169318 
3.5170639 
3 5171959 



3.5173279 
35174J98 
3.5175917 
3 5177236 
3.5178554 



3.5179872 
35181189 
3.5182507 
3299;3.5183823 
3300 351 851 39 



3321 3.5212689 

3322 3 5213996 

3323 3.5215303 
33243.5216610 
3325-3.5217916 



1332 



3.5227050 



3334 
3335 



33333.5228353 



3.5229656 
3.5230958 



3336 
3337 
3338 
3339 
3340 



3.5232260 
3.5233562 
5234863 
3.5236164 
35237465 



8341 
3342 
3343 
3344 
3345 



3346 
3347 
3348 
334? 
3350 



,No. 



,3351 

'3352, 

'3353 

[ 3354i3. 

3355 



Log. 

3.5251744 

35253040 

8.5254336 

5255631 

5256925 



No. 



3401 

3402,«. 

3403*3. 

3404 

3405 



Lo^. 



5316066 
3.5317343 

5318619 
3.5319896 
3.53*1171 



33563 5258220 3406:3.5322446 

3357 3.5259513 3407 3.5323721 

3358 3.5260807 34083.5324996 
!3S59 3.5262100 3409:3.5326270 
13360 3.5263393]3410 3.5327544 



ii *< 



!336l|3.5264685; 3411J3.5328817 



3362(3.52659771 
3363,3 5267269 
3364 3.5268560 1 
3365,3.5269851 



3412,3.5330090 
34133.5331o53 
3414'3.5332635 
341535333907 



33663.5271141 
3367135272431 
3368'S 5273721 
3369.3.527501 Oj 
3370,3.5276299 



3416,3.5335179 
34173.5336450 

3418 3.5837721 

3419 3 5338991 

3420 3.5340261 



3371,3.5277588' 
33723.5278876! 
3373 , 3.5280163. 
3374'3.5281451i 
3375 , 3.528273 8; 
(3376 3.5284024 
■337713.5285311' 
3378'3.5286596! 

3379 3.5287882 

3380 3.5289167 



3421 
3422 
3423 
3424 
3425 



3426J3 5347874 
3.5349141 
3.5350408 
3.5351675 
3.5352941 



3427 
3428 
3429 
3430 



33813.5290452 
3382'3.5291736H: 

3383 3.5293020 : 

3384 3.5293304 I 
13385:3.5295587 ! 



3.5238765! 
3.5240064! 
3.5241364; 
3.5242663! 
3 5243961 



3.5245259 
3.5246557 
3.5247854 
3.5249151 
5250448 



3386 3.5296870 
3387J3.5298152, 
3388 3.5299434 ! 
33893.5300716' 
3390 3.5301997 



33913.5303278 
3392,3.5304553 
33933 5305839 
33943.5307118 
3395, 3.530839 8 

33963.5309673 
3397j3.5310955 
3398.3 5312234 
3399 3.5313512 
3400 ! 3.5314789 



3431 
3432 
3483 
3434 
3435 



3436 
3437 
3438 
3439 
3440 



3441 
3442 
3443 



3446 
3447 
3448 
3449 



3469 
3470 



3.5354207 
3.5355473 
3.5356738 
3.5358003 
3.5359267 



3482 
3483 
3484 
8485 

3.5360532 3486 
3.5361795 3487 



3.5363059 
3.5364322 
3.5365584 



3.5366847 
3.5368109 
S 5369370 
34443.5370631 
3445 3.5371892 



5373153 13496 



3 

3.5374413 
3.5375673 
3.5376932 
345035378191 



3451 
3452 
3453 
3454 
3455 



3456 
3457 
3458 
3459 
3460 



3461 
3462 
3463 
3464 
3465 



No. 



3.5379450 
3.5380708 
3.5381966 
3,5383223 
3.5384481 



8.5385737 
3.5386994 
3.5388250 
3.5389500 
3.5390761 



3.5392016 
3.5393271 * 
3.5394525 
3.5395779 
3.5397032 



3466 3.5398286 
346735399538 
34683.5400791 
3.5402043 
35403295 



3 5341531 3471 3.5404546 
3.5342800 3472 3 5405797 
3:5344069 3473 3.5407048 
3.5345338 3474 3.5408298 
35346606 3475 3.5409548 



3476 
3477 



3479 
3480 



3488 
3489 
3490 



3491 
3492 
3493 



3495 



3497 
3498 



Log:. 



3.5410797 
3.5412047 
34783.5413296 
3.5414544 
3.5415792 



348J&5417040 



3.5418288 
3.5419535 
3.5420781 
35422028 



3.5423274 
35424519 
3.5425765 
3.5427010 
35428^54 



3.5429498 
3.5430742 
3.5431986 



34943.5433229 



3.5434472 



3.5435714 
3.5436956 
35438198 
3499J3 5439439 
35QQI3 543068gl 



ARITHMETICS 



22l 



Log. t|No,, Log. 



No.] 

35^1 3.5441921 3^513.55035071 

3502 3.5443161 '3552 3.5504730 

3503 3.5444401 [3553 3.5505952 
35043 5445641 '35543.5507174 
35051 3.5446880 '3555 3.5508396 
3506*3.5448119 3556 3.5509618 

3507 3.5449358 '3557 3.5510839 

3508 3.545Q596 3558 3.5512059 
3.5451834 3559 3.5513280 
3.5453071 3560 3.5514500 
&.5454o08 3561 3.5515720 



No. 



Log. 



3.5564231 
3.5565437 
3.5566643 
3.5567848 
35569053 



3509 
3510 



3601 
3602 
3603 
3604 

£05 

36063.5570*57 
2607 3.5571461 

3608 3.5572665 

3609 3.5573869 
36103.5575072 



3511 

351213.5455545 ;3562 3 5516939 

3513J3.5456781 ! 3563 3.5518158 

3514'3.5458018 '3564 3.5519377, 

51513.5459253 13565 3.5520595 



No. I Log ._ . , No. 



3651 3.56241 18l|370i 
3652 3,5625308' 3702 

1653 3.5626497^3703 
36543.562768513704 

655 3 5628874 13705 
>656 3.5630002 



611 
3612 
361 
3614 
3615 



706- 

657)3.5631250 J3707 

658 3^632437 3708 

3659 3.56336M 13709 

6603.5631211*3710 



L og. | 



3.5683191 
3.5684364 
3.5685537 
3.5686710 
3.5687882 



3516 ! .3.5460489 t !3566 35521813 
3517 3.5461724 '3567 3.5523031 
351813.5462958; 35683.5524248J 
3519|3.5464193 33569 3.5525465! 
3520 3.546542 7 1 3570 ' 3.5526S8 2| 
3521 3.5466660JI3571 3.5527899J 



36i6 
3617 
3618 



3.55/0275 
3.5577477 
3.5573686 
3.5572881 
3.5581033 
3 5582284 
3.5583485 
3 5584636 
619,3.5585886 



3661 
3662 
366: 
3664 
3665 
3660 
667 
3668 



,620 
1621 
3622 



3.5587086 



3.5689054 
3.5690226 
3.5691397 
3.5692568 
3.5693739 



3.5635997'3711 
3.5637183 ! 3712 
3.5638369 3713 
3.5639555 3714 
3.5640740 ! 5715 



3.5694910 
.5696030 
36697249 
3.5698419 
3.5699588 



o.5ofcti285 

3.5589484 

3.5590683 1 

55918821 



3522|3.5467894 ; J3572 3.5529115! 
35233.5469126:3573 3.5531030113623 

3524 3.5470359, 3574 3.5531545 3624 

3525 3.5471591 3575' 3.5j32760 '!3625 3 5593080 ', 
5526 3.5472823. 3576 3.5533975J 

3527 3.5474055 3577|3 55351 89i 

3528 3.5475286(135783.5536403 

3529 3.5476517 3579 3 5537617 
3530*1 3.5477747 1 3580 1 3.5538830 1 
3531 3.5478977 J3581 (3.5540043 



o.5641925j37l6i.>.o7UU7 57 
3.56-13ld9; 371713.5701926 
_ __, 3.5644293' 1 3718 3.5703094 
3669 3.5645477! 3719 3.5704262 
367ol 3.564666 l! : 3720' " 5705429 
00? 1 13.504? 844 ! : 67~7i |T370659?' 



oO*o|o.5594278 
3627 3 5595476, 



3532 3.5480207p582|3.5541256 

3533 3.5481436 3583|3 5542468 
35343.5482665 35843.5543680 
3535 3.5483894' 
35363.5485123! 



3585 ! 3.5544892 



353713.5486351 
35383.5487578 
35393.5488806 



2672 3 5649027 
3673 3.5650209 
3674^3.5651392 
3675 3.5652573 



00/ o!3 5653755 

3677 3.5654936 

3628)3.5596673' 36783.5656117 

3629 3.5597870. 

3630 3.5599066'j 



,3631 
3632 
3633 
3634 
3635 



3586.3.5546103 
358713.5547314 
3588 ! 3.5548524 

,35893.5549735 

3540 3.5490033! 3590 3 555Q944 



354113.5491259! 3591^ 5552154 
~ 542,3.54924861 3592 3.5553363 



35433.54997121 
3544 ! 3.5494937 
3545' 3.5496162 i 



3593 3.5554572 
35943.5555781 
3595'3.5556989 



35463.54973871 
3547j3.5498612| 
3548 3.5499836 



3636 
3637 
3638 
3639 
3640 
5641 
3642 



.5600262 
3.5601458 
3.5602654 
3.5603849 

J5605044 
3.5606239 
3.560743; 
3.5608627 
3.5609821 



; 561101 4 \3690 



3722 3.5707764 



3723 
3724 
3725 
3726 
3727 
3728 



3.5708930 
3.5710097 
3. 5711263 

5712429 
3.5713598 
3.5714759 

5715924 
35717088 



3679 3.5657297 3729 
36803 .565847 8 3730 ~^ JZZ _ z:zr 

3681 3.5659658 37313.5718252 

3682 3.5660838 3732 3.5719416 

3683 3.5662017 '3733 3.5720580 

3684 3.5663196 3734 3.5721743 
i3685j3.5664S75 ! 3735 3.5722906 



6066 o.506oo5o\jo736|3 5724169 

3687 3.5666731 1 3737 3.5725231 

3688 3.5667909! 3738 3.5726393 



3689 



3.5669087i!37S9|3.5727554 
3.5670264 ! '3740 3.5728716 



3.5612207 |o6yi|3.5671440,3741o\5729877 



3.5613399 3692 



1643 3 5614592 

3644 3.5615784 

3645 3.5616975 



3596 3.5558197 U646 3.5618167 
3597,3.5559404 3647 3.5619358 
3598 3.5560612 3648 3.5620548 



369: 

8694 
,3695 
3696 
13697 
3698 
3699 



3549 3 5501060 3599 3.5561818 3649 3.5621739 
3 55013.5502284113600 3.5563025'13650l3 5622 929.l3700 

29 



3.5672617i!3742 3.5731098 
3.567«793i ! 3743 35732198 
3.5674969; 3744 3.5733358 
5676144 ! ! 3745 3.573451 P 
0.66779*0 S746 3"5?35S7£ 
3.5678495:' 3747 3 573683T 
3.5679559|374* 1 } 5737996 
3.5670843: 3741 h. 5739 154 
3.56720171 3750!3.57403li: 



i 



22S 



LOGARITIIMICK 



37513. 
37523 
37533 
375^3. 
37553. 



i\i>[ Lotf. 



37563 
3757:3. 
3758'3 
3759)3. 
37603 

3761*3 
3762J3 
37633 
3764 3 
3765:3. 

3766|3 
3767 3 
37683 
376913 
37703. 

6JYI 
3772 
3T73 
3774 
3775 
3776 
•J777 
3778 
3779 
3780 
3781 
3782 
3783 
3784 
3785 



.6741471 

5742628! 

5743786 

574494? 

5746091 

5747$5( 

57484U 

574956h 

575072.5 

5751878 

5753099 

5754188 

575534$ 

5756496 

5757650 



ooi/l 3.579#97< 
3802,3 5800121 



3.5801263 
3 580240^ 
3 5803547 
£5804688 



380: 

3804 

3805 

'3806 

3807 

3808 

3809 

3810 

38li!oj«10389 



3.5805821 3857 3.5862496 



3 5806969 
3 5808110 
3 5809250 



5758803 

5759^56 

.5761109 

5762261'' 

.3763414 

5764565 

5765717 

5766868 

.5768019 



.\«i.» i^^. 



'3Hd1o.o&5o735 
138523.5856863 
•3853 3-5857990 
38543 5^59117 
3855 3J860244 
ooob o. 5861 370 



3901 
3902 
3903 



3 6911760 

35912873' 

3.5913986 

3904 , 3.5915098 

3905J3 5916210 

906'vi o9l7922 

3907^3-5918434 

3908 3.5919546 

3909 3.5920657 
3910J 3 5921768 

^Sol o^Loo^6i!39n'3 5922878 
381213 5811529(!3862 3.5868123 ]39123 5923988 



.38583 5863622. 
J3859 3 5864748: 
^60 3 58658731 



l^og. h^u.j Loj>. 



3951 OO967070 
'3952 3 5968169 
]3953 3 5969268 
,39543.5970367 
3955J3 5971465 



o. ob If 2563 
35973661 
3.5974758 
3.5975855 
3 5976952 



3863 3.5869247 13913:3.5925098., 

3864 3.5870371 ! 3914 3.5926208 

13865 S_5871495|i3915!^5927318 

13866 5.5872618| 1 ;*16 ! 3*5928427 
|3867 : 3.587S743| i 39l7 , 3 5929536 
13868 3 5874863 39183.5930644 



3813 35812668 
38U3.581S807 
3815 3 581494 5 

J3816J.5816084 

! 3817'3.5817222 

,3818 3.5818359,-- . „ tnrt ,_ 

*3819 35819497 3869 3.5875987 |j39 19 3.5931753 



3822'3.5822907 3872 3.5879751 

3823 3.5824043 ! 3873-3.5880475 

3824 3.5825179 13874 3.5881596 
5769170 38253 5826314 J3875 3.5882717 



.>,86 

3787 

37883 

3789 

3790 



3791 

3792 

3791 

3794 

3795 

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3797 

37983. 

3799 

3800 



8203 5820634 J3870 3.5877110 



3821 6o»21770 



j33n;3.5878232i!3921 3 5933968 



J3920 3 5932861 



.5770320 38263.5827450 
.5771470 38273.5828585 
.5772620 38283.5829719 
5778769 38293.5830854 
5774918 3830 3.5831988 '3880 3.5888317 ; 

5776097 o<\>1 3.5833122 
5777215 38323.5834255 
5778363 38333.5835388 
5779511 38343.5836521 
5780659 38353.5837654 
5781806 otto6 3.5838786 
5782958 38373 5839918 
•5784100 38383.5841050 
.5785246 38393 5842181 
.5786392 38403.5843312 



l3925 ; 35938397 



;3o; (i3 5883«38, 
138773.5884958 
'38783.5886078 
138793.5887198; 



.5/87538 3841 3.5844443 
5788683 38423.5845574 
5789823 38433.5846704 
579397S! 38443.5847834 
792118133453.5848963 



5799262: 38463.5850093 
.5794406 f 3847 3.5851 222 
5795550, 38483.5852351 
.5796693 38493.5855479 
5797836 38503.5854607 



3926 o.o K Jb^o03 

927 3.5940509 
3928/3 5941715 
3929 3.5942820 

930*3 5943926 3<)80 35998831 



; 3881 3.586943d 
38823.5890555 
! 3883 3.5891674 
'38843.5892792 
3885 3.5893910 



3bb63.589502tf 
38873.5896145 
38883.5897263 
38893 5898379 
389035899496 
'3891 o.oy00G12 
3892^.5901726 
3893 3.5902844 
38943.5903959 

3895 3.59050 75 

3896 o^uol89 
38973.5907304 
38983 5908418 
38993 5909582 
39003.5910646 



3 5979145 

. 3.5980241, 

13964 35981336 

3965 3 698243% 

3966 

>967 

3968 
;969 



o.^b3327 

3.5984692 

3.5985717 

._, 3.5986811 

3970 3.598 7905 

^71 

1972 

3973 



(3922,3.5935076 

8923'35936083 

3924t3.5937290 39743.5992279 



o.o*,t&999 
3 5990092 
3.5991186 



3975 
3976 
3977 
3978 
3979 



3 5993371 
3.5994464 
3.5995556 
3.5996648 
35997739 



o93lM-^45030 
3932'3.5946135 
3933-3 5947239 
8934*3.5948344 
3935? 3 5949447 
3936i3.5950551 
3937|3 5951654 
3938'3.5952757, 
3939;35953860 
3940 3 595496 2 
3941 o 5956064 

3942 3.5957166 

3943 35958268 

3944 3.5959369 

3945 3^960470 

3946 3 5961571 
39473.5962671 
3948 3.5963771 
39493.5964871 
39503.5965971 



,i>joi o.jb>99922 
|3982|36901013 
3983J3.6002103 
3984'36003193 
3985* 3 6004283 
3986'3.6005o/3 
3987|3.6006462 
3988i3.600755\ 
3989 ! 3 6008640 
399036009729 



3991J.ooiubl7 
39923.6011905 
9933.6012993 
39943.6014088 
399*3 6015168 



3 ( >9bo.uv/ 10255 
3997'3 6017341 
39983.6018428 
3999 3.6019514 
40003 6020600 



ARITHMETICS 



$23 



400 i 
4002 
4003 
4004 
4005 



;no. 



Lo»{. J 



5.602 lotft 
.0022m 
3.6023857 
3.6024941 
3.6026025 



3 6076694 
4055 J 6077766 

4054 3.6078897 

4055 3-6079909 



iO06*5, 
40073 
4008'S, 
50093; 
4010'3 

40Tl|3 
4012'3 
4013 3 
40143 
4015*5 



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6028193 
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,6052ju7 , j 
6033609 
6034692 
6055.T4 1 
6036855' 



4061 3 60863 JO 4111 
4062 3.6087399* 4112 36140531 
4063 3.5088468 '4113 36141587 
40643.6089537 41143 6142643 
4065|3 6090605 41153.6143698 



40l63.'J03fyJ7, 
40173 6039018 
40183.6040099 
40193.6041180 
402036042261 



4066' 
4067 
4068 
4069 
4070 



4JJ1 3.604JJ41 
14022 3 6044421 
4023 5.6045500 
40243.6046580 
4025 3 6047659 



40:40 3.6048738 
40273.6049816 
40283.6050895 
40293.6051973- 
403036053050 



4031S6054128 

4032 3.6055205 

4033 3.6056282 
40343.6057359 
4035 3.6058435 



4^56 36029512 
4037 3.6060587! 
40383.6061663' 
4039 3.6062739 
40403 6063814 



40-* i 3.60d4«ay 1 
40423 6065963 
40433.60670371 
40443.6068111! 
4045 3.6069185 



40463.607025* 
40473.607135$ 
40483.5072405 
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40^)3 6074550' 



\u. 



4051 
4052 



Loir. 



£so.\ Lo 



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4056 5.6080979J 



4057 
4058 
4059 



3.6082050 
3 6083120 
3 6084190 
3.6085260; 



4101 3.6128898 

4102 3.6129957 



410, 
4104 
4105 



3.6131015 
3.61^2074 
3 6133132 



410b 
4107 
4108 
4109 
4110 



O.UU910/4 4.116 
3.6092742|4117 
5.6093809 14118 
3 6094877 J4119 
3 6095944 4120 



415613 
415713 
4158;3 
4159,3. 
4160;3 
.61394/5 |4161!3. 



.61341 8y 
;.S1S5247 
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I.6137361J 
I 6138418 



NiO. 



4151 
4152 
415: 
4154 
4155 



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3.6181527 
3 6182573 
3.6183619 
3.6184665 
3.6185110 



|4162| 
1416313, 
4164J3 
4165I3 



3 6147918 
6148972 



4071J5.609/011 
4072 3*6098078 
4073^3.6099144 
40743.6100210- 
4075J3. 610127 6 

40763?61u2342 
40773.6103407 
.6104472 
3.6105537 
3.6106602 



412113.6150026 
4122l3.615108G J 
41233.6152193 



4124 
4125 



4078 
4079 
4080 



4081 3.6 10? 666 
4082 3.6108730 
408336109794 
4084'3.61 10857 
408513.6111921 



4128 
4129 



1.61-53187 
1.6154240 



4126 3.6155292 

4127 3.6156345 



6186755 
6187800 
6188845 
6189889 
6190933 

6191977 
6193021 
.6194064 
.6195107 
.6196150 



4206 3.0238693 
42075.6239725 
4208 3.6240757 
42093.6241789 
42103.6242821 



3.6144754 4 1 6613.0197 193 
3.6145809 4167|3.6198235 
3J5146863 I416S3.6199277 
: 4169 5 6?00319 
! 41703 6201361 



41715.6202402 
41723.6203443 
141733.6204484 
, 4174'3.6205524 
14175^6206565 



41763.6207605 

41773.6208645 

3.6157397141783.6209684 



3 61584491 



4130 3 6159501 



4131f3.61605.52j 
41323 6161603J 
4133 3,6162654 
4134*3 6163705, 
4135*3 6164755J 



40«6j3.61 12984 41363.6165805 

4087 3.6114046 J4137J3.61 66855: 

4088 3.6115109 41383.6167905, 
4089j3 51 16171 4139!3 6168954? 
4090^36117253 41 40^5 61 70003 ! 



409115.6118295 
409213.6119356 
4093'3.6120417 
4094|3.6121478 
4095 3:6122 »39 



^i . 



4141 3 
!4142 3 
4143J3 

41443 
141453 



6171050 
6172101 
6173149 
6174197 1 
6175245 



4096 
4097 



4099 



3.6123599 4146.7 
3 6124'660 14147|3 



6176293; 
6177540; 
6178387! 



4098 3 6125720 ;4148j3: 

5.6126779 |4 14913. 6179434 1U99 
150' 



4100'3.6127839> 



J.6180481 



41793 6210724 
4180 3 6211763 



4181 '36212802 
4182.36213040 
4183 3.6214879 
41843.621591 
4185 36216955 



41863.6217992 
4187j3.6219630 
4188,36220067 
4189 f 3.6221104 
4190 3 6222140 



41913. 
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4195;3. 
1.1943. 
4195 ! 3. 



.6223177 
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6V2732C 



4196,3. 
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4198 3. 



4200: 



.6228354 | 

62293SC] 

6230424 

633145S 

6232493 



No. I Log. 



4201.3.6233527 

4202 3.6234560 

4203 3 6235594 
42043.6236627 
4205 3 6237660 



42113.5245852 
421236244884 
4215 5.6245915 
J4214 5.6245945 
42153 6247976 



;4216 5o249006 
142173 6250056 
14218 5.6251066 
'4219 5.6252095 
42205 6253125 



4221 5.0254154 
42225.6255182 
4225S.62562I1 
J4224 56257239 
J4225 56258267 



42263. 
42273, 

4228 3 

4229 3. 

4230 3 



6259295 
6260322 
6261350 
6262377 
6263404 



4231 
4232 
4235 
4234 
4235 



6264430 
6265457 
6266483 
,6267509 
,6268534 



42363 6269560 
4237 3.6270585 
;4238 , S6271610 
|4239!3.6272634 
4240'3,627S659 



,424 1 ! 3. 6274685 
12423.6275707 
142433 6276730 
i4244 3 6277754 
4245 3 6278777 



42463.0279800 
•424736280820 
14248 36281843 
42493.6282865 
142503.6283887 



tu 



LOGARITHMICK 



4Jd1'3 6284911 
6285933 
6 86954 

6087975! 
^2SS996* 



42J23 
4253 
4254 
4255 



425o<3 
42573 
4253*3 

4259|3 
4260:3 

42bl!3 
4262|3 
4263,J> 
4264|3 
4265J3 



N T o.| Log. 



,Ncj 



4301J3 

43U2|3 
4303'3 
43043 
4305!3 



\ k«g || No. Lo^. 



O^jjaloj43u63 
629l03r!4307 r i3 
629^05^43083 
6293076 U309'3 
629409t>!;4310'3 



6335694 4351 3 
63367<M , !43523 
6337713|43533 
63387231435412 
6339732:14355 3 
6j40/4^|4o563 
6341749; |4357 3 
6342757! |4358 3 
6343765; J4359 3 
6344773 ]4360 3 



4266P 
42673 
42683 
42693 
42703 



t>4/511jj:4311;3 

629bl34 14312 ! 3 
629*153 ;4313'3 
6298172 *4314 ! 3 
6299190f.4315 , 3 



431t>j3 
4317& 
4318;3 

143193 
143203 



6o*57bU ;43bl 3 
6346788 4362 3 
6347795 4363 3 
6348801' 4364 3 
63498084365 3 



6385891 
6386889 
6387887 
6388884 
6389882 



63908? 9 
6391876 
6392872 
6393869 
6394865 



0395861 
6396857 
6397852 
6398847 
6399842 



No.| Log. 



44Ul;3 6435d1* 
1402 3 6436500 



4403 



44043 6438473 



4405 



4406 
4407 
4408 
4409 

14410 



6300^09 
6301226 
6302244 
6303222 
6304279 



4271|3 
42T23 

4273p 
42743 
42753 



6^05290 
6306312 
6307329 
6308345 
6309361 



4276,3 
42772 
42783 
427913 
4280|3 



631037/ 
6311393 
6312408 
6313423 
6314438 



4281|3 
4282'3 
4283'3 
4284 ! 3 
4285'3 



6315452 
6316467 
6317481 
6318495 
6319508 



4'itf 6 3 
4287:3 
42883 
4289^3 
42903 



6J2J522 
6321535 
6322548 
6323560 
6324573 



42913 
42923 
42933 
4294*3 
4295;3 



6325585 
6326597 
6327609 
632*620 
63295.V2 



42963 6330643 
429713 6331654 
4298'3 6332664 
4299:3 633367* 
4300'3 63346* 



6350814 
6351820^ 
6352826' 
6353832! 
6354837 



4321 
4322 
4323 



43243 



63558431 
6356848 
6357852 f 
63588571 
63598611 



43263 
43273 
43283 
43293 
43303 



6360805 
6361869 
6362873 
6363876 
6364879 



'433113 
143323 
143333 
, , 4334 , 3 

J4335I3 



6365882 
6366884 
6367887 
6368889 
6363891 



43661, 
4367 
4368 
4369 
4370 



4371 

4372 

4375 

43743 

43753 



4376 
4377 
4378! 
4379 
43803 



4381 
4382 
4383 
4384 
4385 



4336|3 6370393*43863 
433713 6371894 J4387 
4338J3 6372895 '4388 
4339 3 6373897 '4389 3 
4340|3 6374897:4390 3 



4341 
4342 
4343 
4344 
4345 



43463 

4347 

,43483 

434913 
14350 



6375898 
6376898 
6377898 
6378898 
6379898 



•6380897 
6381896 
6382895 
6383894 
638489: 



;4391 

14392 

4393 

4394; 

4395 



4396 
4397 
4398 
4399 
4400 1 . 



64UU837J 
6401832| 
6402826! 
6403820J 
6404814 



4416'3 
44173 
44183 
44193 
4420J3 



6405808 
6406802 
6407795 
6408788 
6409781 



6410773 
6411765 
6412758 
6413749 
6414741 



64157331 
6416724 
6417715 
6418705 
6419696 



6420686 
6421676 
6422666 
6423656 
6424645 



6425634 
6426623 
6427612 
6428601 
6429589 



6430577! 
6431505 
6432552 
6433540 



4411 

4412 

4413 

44143 

4415! 



3 643748; 



, No. ' 

4451 
14452 

445. 



6139459 



6440445 
6441431 
6442416 
6443401 
6444386 



Log- 



3 6484576 
3 6485552 
3 6486527 
4454 3 6487502 
44553 6488477 



6445371 
6446955 
6447399 
6448323 
6449307] 



44613 
4462*3 
4463 ; 3 
4464'3 
4465',3 



6450291 
6451274 
6452257 
6453240 
6454223 



4421|3 
44223 
44233 
44243 
44253 



6455205 
6456187 
6457169 
6458151 
6459133 



44263 
44273 
44283 
4429(3 
4430*3 

4431-3" 
44323 
4433 3 
4434 ! 3 
4435J3 



6460114 
6461095 
6462076 
6463057 
6464037 



4466;3 
44673 
4468*3 
44693 



44713 

44723 

4473 

44743 

4475 



6465018 
6465998 1 
6466917^ 
6467957 
64689361 



44363 
4437J3 
44383 
443913 
4440 3 

44413 
44423 
44433 
44443 
44453 



6469915! 
6470894! 
6471873J 
6472851 
6473830 



6474808 
6475785 
6476763 
6477741 
6478718 



44463 
444?;3 
44483 
4449 3 



3 643452744 50 3 



6479695' 
6480671! 
6481648 
6482624 
6483600! 



44563 6489452 
44573 6490426 
44583 6491401 
4459 3 6492375 
44603 6493349 



6494332 
6495296 
6496269 
6497242 
6498215 



6499187 
6500160 
6501132 
6502104 
6503075 



44763 6509901 
44773 6509871 
44783 6510841 
6511811 
44803 6512780 



4481 
4482 
4483 



4489 
4490 



4498 
4499 
4500 



6504047 
6505018 
6505989 
6506960 
6507930 



3 6513749 
3 6514719 
3 6515687 
44843 6516656 
44853 6517624 



44863 6518593 
44873 6519561 
44883 6520528 
3 6521496 
3 6522463 



4491 3 6523451 

4492 3 6524397 

4493 3 6525364 
44943 652633V 
4495 3 6527297 



4496 3 6528263 

4497 3 6529229 
3 6530195 
3 6531160 
3 6532125! 



ARITHMETICK. 



2ft 



4501 
4502 
4503 
4504 
4505 



450613.6547912 
450*3.6538876 
45083.6539839 
36540802 
3.6541765 



4509 
4510 



4511 3.6542726 
45123.6543691 
451313.6544653 
4514 , 3.6545616! 
45153,6546578' 



4516<3.6547o39! 
4517&654850li 
45183.6549462: 
4519,3.6550423 
45203.6551384 



No 



3.6533090 
3.6534055 
3 6535019 
3.6535984 
3.6536948 



Lo£.- 



3.6695359 

3.6596310 

45683.6597261 

,6598212 

3.6599162 



4569 
4570 



4121(3.6552345 
45223.6553306 
45233.6554266 
4524'3.6555226 
45253.6556186 



45263.6557145 
4527J3.6558105 
452836559064' 
4529,3.6560023! 
45303.6560982' 



45ol3 6561^41 
4532 3 6562899 
4533-36563857. 
4534 f 3 6564815 
453536565773 



4536 3. 
4537|3 
4538.3 

45393, 
4540 f 3 



65667^0 
6567688 
6568645 
6569002 
6570559 



4541,3. 
45423. 
4543i3 
4544|3' 
4545 3. 



6571515 
.6572471 
6573427 
6574383 
.6575339 



45463. 
4547,3 
4543,3 
45493 
4550'3 



,65762y4i 
6577250,' 
6578205! 
6579159 
6580114 



No. Lop:. 



4551 
4552 
455^ 
4554 
4555 



3.65810* 

3.6582023| 

3.6582977 

3.6583930J 

3.6584884! 



4556 
4557 
4558 
4559 
4560 



3.6585837 
.6586790 
3.6587743 
3.6588696 
3.6589648 



4561 
4562 
4563 
4564 
4565 



3.6590601 
6591553 
3.6592505 
3.6593456. 
3.6594403 



4566 
4567 



4571 

4572 
4573 
4574 
4575 



4577 
4578 
4579 
4580 



4581 
4582 
4583 
4584 
4585 



4586 
4587 
4588 
4589 
4590 



4601 

4602 

4603 

4604)3 

4605 



4606 
4607 
4608 
4609 
4610 



4611 
4612 
461, 
4614 

4615 



3.6600112 
3.6601062 
3-6602012 
3.6602962 
3.6603911 



4616 
4617 
4618 
4619 
4620 



4621 
4622 
4623 
4624 



4576 3 66048OU, 

6605809i 

3.6606758 

3.6607706; 

3.6608655; 



3.6609603 
3.6610551 
3.6611499i 
36612446 
3.6613393 



3.6614341 
3.6615287, 
6616234 
3.6617181 
3.6618127 



4591j3.6ol9u/o 
4592j3.662001y 
4193J3.6620964, 
45943.6621910 
4595I3.6622855 1 



45ybj3.6623800 
45975.562*745! 
3.6625690 
3.6626634 
3.662757& 



4598 
4599 
4600 



No, 



3.6637951! 
3.6638893 
3.6639835 
3.6640776; 
3.6641717 



3.6642658 
1.6643599' 
3 6644539! 
3.6645480 1 
36646420' 



4625 3 



4627 
4628 
4629 
4630 



4o31 

4632 
463: 
4634 
4635 



4636 3~ 
4637 

4638 
4639 
4640 



Loir. "| f No. | Log. 



3.6628522 
3.6629466 1 
3.6630410J 
,6631353| 
3.6632296 



4651 3.6675463 
46523.6676397 
4653 3.6677331 
46543.6678264 
4655 3.6679197 



3 6633239 4666 3.6680130 
3.6634182 46573 6681062 
3.6635125 4658 3 6681995 
3 6636 J67i 4659 3.668292 
3 6637009 4660 3.6683859 



3.6647360' 
3.6648299, 
3.6649239! 
3.6650178 
.66511171 



4626 3.6652056 
3.6652995! 
3.6653934] 
3-6654872j 
3.6655810' 



3 6656748; 
36657686 
3.6658623 
.6659560 
3.6660497, 



6661434 

6662371! 

3 66633071 

3 6664244; 

6665180 



,6666116 
6667957j 
66679871 
6668922 
.6669857; 



.6670792' 
3#671727 
3.6^72661! 
3.6673595' 
3 6674530- 



46613 

4662 3. 

4663 3. 

4664 3, 
46653 



No- 



4701 
4702 
4703 
4704' 
'4705 



Lotf- 



3.O721903 
3.6722826 
3.6723750 
3 6724673 
3 6725596 



47 06 
.4707' 

'4708 
4709 
4710, 



6684, Jyl* 
6685723 
6686654* 
6687585 
6688316 



46663 
4,6673, 
46683, 
46693 
46703, 



4/11! 

4712 1 
4713 1 
4714' 

4715 ; 



.6681)447 
66903? 8 
6691308, 
6692239 ; 
66931691 



4/16 

4717» 
4718' 
4719 1 
4720 1 



46713 6694099 
46723.6695028 ! 
46743.6695958! 
46743.6696887; 
4675 3 669781 6, 

46763.669874'5 
46773.6699674! 
46785.6700602 
46793.6701530 
4680 3.6702459 



4/21 
4722 

4723! 
4724 ! 
4725: 



4681 3.6703386 

4682 3.6704314 

4683 3 6705242 
4684'3.6706169 
4685*3.6707096 



4686 3.6708023 
4687'3.670S950 
46883.6709876 
4689'3.6710802 
46903 6711728' 



469 1,3.67 126o4l 



4692 
4693 
4694 
4695 



6713589 
3.6714506^ 
3.671543-11 
3 6716356' 



4696 
4697 
4698 
4699 
4700 



3.671/281! 
3.6718206' 
3 6719130' 
3.67200541 
367209791 



4726i 

4727J 
4728 
4729! 
47301 



4/31 
4732 
4733 
4734 
4735 



o o/265ii.' 
3.6727442 
3 6728365 . 
3.6729287 
3.6730209 



3.6/31131 
3 6732053 
3 6732974 
3.6733896 
3 673481 7 
o.o/Jj/38 
3 6736659 
3.6737579 
3.6738500 
36739420 



3.6740J40 
36741260 
3.6742179 
3.6743099 
3 6744018 



J.6744937 
3.6745856 
3.6746775 
3.6747693 
3.6748611 



3.674^529 
13.6750447 
3 6751365 
13.6752283 
l3.fi/53200 



4/jb 
4737 
4738' 

47391 
4740; 



o.u/ 04117 
: 3.67 55034 
3 6755951 
3.6756867 
6757783 



4741' 
4742; 
4743; 
4744i 
4745 1 



4746> 
4747! 
4748 
4749 
4750 



o.o? 58700 
3.6759615 
3.6760531 
3.6761447 
367*2362 



a.o/t)3277 
3.6764192 
3.6765107 
3.6766022 
36766936 



526 



LOGARITHMICK 



No. L 



|N«>., Leu* 



47oi 3.0/orojj 
47523.6768564 
4753 { 3 6769678 
4754 3.6770592 
4755'3.6771.505 



<i 8 Ji 3.68133 1/ 
4802 3.6814222' 
i4803'3.6815126 
! 4304'3.68t6010 
48053.6816934 



47*>o 2, 
4757% 
47583 

4759 \S. 

4760 3. 
4701 3" 
4762 3, 
4763;3. 
4764'3. 
476.^3. 



,j//:**i6 l A&/b3. 
,5773332^4807'3 
6774244 1 4808,3 



677515/; 

6776070' 



o77ui/o<; 
.6777894 
,6778806 
6779718 
,6780529 



4/uo3.o7t>io4u 
4767,3.6782452 
47683.6783362 
4769:3.6734273 
47703.6785184 



4/71 '3.67*01/** 
4772J3.5787004 
4773J3.6787914 
47743.6788824 
4775'3.67897S4' 



47/63.57yoo4j 
4777|3.6791552 
4778,3.6792461 



47803.5794279 



480$: 
4810,3 
48113" 
48123. 
481313, 
48143 
4815 3. 



4851 b 6858313 
4852*3 6859208 

4853 3.68601C3 

4854 ;3 .6860998 

4855 3.686189 2 

6817838 !48o6:3.6802/«7 
.6818741 4857'3.68f>3681{ 
6819645! 4858*3 6864575J 
6820548 1 48&I3.6865469 1 
6821451' 4860'3 6866363 



.O8^oo4 
6823256 
.6824159, 
6825061, 
6825963 



4862 
4863 
4864 
486 



481o3.6826865 
48173 6827766 
4818 3.6828668 
48193.6829569 
48203.6830470 



4821 3.6831371 
48223.6832272 
4823 3.6833173 
48243.5834073 
48253.6834973 



4861 '3-6867256 



4866 
4867 
4868 
4869 
4870 



4826 3.6835873 
482736836773 
148283.6837673 



**/ f o,j.os y&kOl fbojo j.cvwfo/** 

4779 3.6793370 14829 3.6838572 



'48303.6839471 



4871 3.68761811 4y21;3.by20534|!4971 

4872 3.6877072!i 4922 3.6921416! 4972 
487S 3.6877964 4923 3.6922298! 4973 

4874 3.6878855, 49243.6923180 4974 

4875 3.68797461 4925 3.6924062 '4975 



487636880637 
3.6881528 
3.6882418 
3.6833308 
36884198 



4877 
4878 
4879 
4880 



4781 3.679518/1.4831 3.6840370 14881 

4782 3.6796096 4832 3 6841269 4882 

4783 3.6797004 4833 3.6842168 '488. 

4784 3 6797912 '4834 3 6843066 ,4884 
4785 3.679981 9) j 4835 3.6843066 [ 4885 

4786 3 6799/27! 14836 3.6844863 ,4886 

4787 3.6800634 4837 3.6845761 4887 
47883 6801541 4838 3 6846659 '4888 

4789 3.6802448 !4839 3.6847555 '4889 

4790 3.6803355 484013.6848454 4890 



4/yi 3 68042o2 
1792 3.6805168 
47933.6806074 
47943 6809680, 
1795 3.6807836! 

4796 3 6808792 
47973.6809692 
47983 6810602 
47993.6811507 
4800 3.6822412 



4841l3.68493ol 
48423 6850248 



4843 
4844 
4845 



4846 
4847 



4849 
4850 



3 6851145 
3.6852041 1 
3.6852938 



4891 
4892 
4893 
4894 
4895 



3 6853834; 
3.6854730; 
'48483.6855626^ 
3 6856522J 
3.6357417 1 



I 



48' 

48< 

4898 

|4899 

4900 



3 6868150 

6869043 

3.6869936 

3 6870828 



3.6871721 
3.6872613 
3.6873506 
3.6874398 
5.68'5290 



No. f L,og:. i!No| Lotr. 

49U1 3.by02847U951 3.6946929 

4902 3.6903733 Ij4952'3.6947806 

4903 3.6904019 |4953 3.6948683 



49043.6905505 
4905 3.6906390 



49543 6949500 
f4955 3 6950437 



4906 3.090727 j, ,4956 o. 
4907:3'6908161;;495713. 
49083 6909046:4958 3. 

4909 5.-6909930! 14959,3. 

4910 3 



49fcl|3. 
4962J3. 



6951313 
6952189 
6953065 
.6953941 
6954817 



5.69U699j 
L6912584 1 
5.6913468! 



149633, 



3.6885088 
3.6885978 
3.6886867 
3.6887757 
3.6888646, 



3.6889535 
3.6890423 
3.6891312 
3-6892200 
36393089 



3.6893977 
.6894864 
3.6895752 
3.6896640 
36897527 



3 6898414 
3.6899301 
36900188 
16901074 
3 6901961 



4911 3, 
49123 

4913 3 ,ww. i -«w W w. 

4914 3.69143521 14964 !3. 

4915 3.691523 5J!4965'3 
49163.6916119 
4917'36917002 
49183.6917885 



6955692 
6956568 
6957443 
.6958318 
6959193 



J4966J3 6960067 
'4967I3-6960942 

l4968'3.6961816 

4919 3.6918768 J4969J3.6962690 
49203.6919651 l 4970!3.6963564 



4926 3 

49273. 

4928i3, 

4929 3.6927588,(4979 

4930 



3.6964438 
3.6965311 
3 6966185 
3.6967058 
3 6967931 



.6924944:14976 
,6925826! 4977 
6926707(4978 

5.6928469,4980 



J4931 3.6929350 J4981 

4932 3693023i;'4982 

4933 3.693UUIJ4983 
49343.69319914984 



[4935 3 6932872 14985,3 6976652 



>936 
|4937 
4938 
4939 
4940 



4941 
4942 
14943 



4946 
4947 



6968804 
36969676 
3.6970549 
3.6971421 
3. 697229 3 
3 6973165 
3.6974037 
.6974909 
36975780 



3.693*752 J4986 3.6977523 
36934631' 4987'3.6978394 
3.693551 1 J4988 3.6979264 
3.6936390 J4989!3 6980135 
3.6937269 1 4990 3 698100 5 

|4991 r3.6981876 
4992j3.6982746 

4993 3.6983616 

4994 3 6984485 

4995 3.6985355 



3.6938149 
6939027 

3.6 39906' 
49443.6940785 
4945 3.6941663 



3.6942541 
6943419 
494813 6944207 
4949 3.6945174 
'4950 3.6946052 



4996U 6986224 



499f 
4998 
4999 
5000 



3.6987093 

6987963 

3.6988831 

16Q89rOO{ 



ARITHMETICS. 



227 



^\o 



5001 
5002 
5003 
5004 
5005 



50213.70079U2 
5022)37008762 
5023J3.7009632 
5024^.7010496 
502513.7011361 



iuuo 
5007 
5008 
5009 
5010 



5013 
5014 
5015 



5017 
5018 



Lotr. 



3.6990569 
6991437 
3.6992305 
3 6993173 
3.6994041 

3.6994908 

6995776 

6996643 

3 6997540 

3 6998377 



50113.6999244 
50123.7000111i 
3.7000977 
3.7001843 
3.700270'. 



501637003575 



3.7004441 
3 7005307 
5019J37006172 
5020370070S7 



5026(3.7012225 
502713.7013089 
5028|37013953 
50293.7014816 
5030.37015680 



50313.7016543 
5032;3 7017406 
5033 3.7018269 
5034J37019132 
5035'3.7019995 



50363.7020837 
5037(3.7021720 
503&3 7022582 
5039J3.7023444 
5040J 3 7024305 
50413 7025167 
50423.7026028 
50433.7026890 



5044 
5045 



5046 



5048 
5049 



.7027751 
17028612 



No. | Log. 



J No. i Lou. 



505153.7033774 
505213.7034633 
5053;37035498 
50543.7036352 
505513 7037212 



50563.7038071 
50573.7038930 
50583.7039788 
50593.7040647 
50603.7041505 

5061|3.7042263 
5062|3.7043221 
50633.7044079 
5064 3.7044937 
50653.7045794 



i5101 37076553] 
51023.7077405; 
51033.7078256 1 
51043.7079107! 
5105 3 7079957! 



iNtj.f Li>y. | 

5lll3.m«yl5 
5152:8.71 19759 
5153 3.7120601 
5154B7121444 
51 55J3 7122287 



50/1 
5072 
5073 
5074 
5075: 



5066|3.7046653 
3.7047509 
50683.7048366 
50693.7049223 
3.7050080 



.7050936 
.7051792 
.7052649 
.7053505 
.7054360 



5076(3. 

507713. 
507813. 
5079\3. 
50803. 



7055216 
.7056072 
705692/ 
,7057782 
7058637 



506113. 
5082)3. 
5083'3. 
5084b 
5085:3. 



7059492 
7060347 
7061201 
7062055 
.7062910 



5086-3. 
508713 
5088 l 3. 
5089:3. 
50903 

50911 
50923 
5093 3 
50943. 
50953.: 



,7063764 
7064*17: 
7065471 
7066325 
7C67178 



70 8031 
7068884 
7069737 
7070589 
7071442. 



50473 7030333 



3.7029472! 50963 7072294 
5097 3 7073146 
5098;37073998j 
5099J3 7074850! 
5050^ 7032914 1 5100)37075702' 



3.7031193!-: 

37032054 



51063.7080808 
510737081659 
510837082509J 
5109,3.7083359 
511037084209 



511137Ub5059. 
511237085908! 
5113 37086758 
511437087607J 
51l5j370ii8456 



alid'S 71*312! 

5157 8.7123971 

5158 37124813 
5159.37125655 
5160 3 7126497 



5116!o.708y305 
511737090154 
5118 , 37091003 l 
5119 ! 3 7091851 
512037092700 '5170 



5ltol 
5162 
5163 
5164 
5165 
6166 
5167| 
5168 
5169 



L\u| L(»jr. 



o20l 3v 100869 
520237161703 

5203 37162538 

5204 3 7163373 

5205 37164207 



;c6 3.7 166042 
5207 3 7165876 
520a 37166710 

5209 3.7167544 

5210 37168377 



3 /U,3o9\ 5£ll o71o92.il 
3.7128880! 5212 3 7170044 
'3 7129021 '5213 37170877 
3 7129862 J52U 37171710 
3 7130703.I5215 3.717254: 



5121 3.7093548 5171 o.< io5745 
5122J3 7094396 j5172 37136585 
51233709524415173 37137425 
5124'3,709609: 5174 37138264 
512537096939 ;51 75 3 7139104 



5126 3.7097786 
5127(37098633 
512837099480 
5129,37100327 
513037101174 



513137102020 
513237102866 
5133 37103713 
513437104559 
51353 7105404 



513657106259 
513737107096 
51383 7107941 
513937108786 
514037109631 



_ . 13-1 544 (a* 16 3.71/3376 
J37132385 J5217 37174208 
3 7133225 J5218 37175041 
37134065 15210 37175873 
3 7134905 5220 3, 7176705 

0^13./ 177537 
|5222 37178369 
5223 3 7179200 
J5224 37180032 
'5225 37180863 



5176 3 7139943 
'5177 37140782 

5178 3 7141620 

5179 37142459 

5180 37143298 

518137144136 

5182 37144974 

5183 3 7145812 

5184 37144665 
51853 7147488 



5186 3./ 148325 
:5187J37149162 
J5188 37150000 
5189137150837 



5190 

5i41371l0476j:5l9l 
5142 3 7111321 5192 
51433 7112165J15193 
51443,7113010^5194 
5145 37113854 : 5195 

51463 71 1469* ; 5196 
51473 7115542 5197 
5148 3 7116385 519P 
51493 7117229"5199 
51503 7118072i'520C J 



37151674 



o/15^10 
3 715334; 
37154183 
37155019 
37155856 



o./ 136691 
37157527 
37158363 
3 715919^ 
37160033 



o226 3./1616y4 
5227,37182525 
6228 37183356 
5229 3 7184186 
=5230 3 7185017 



5231 3 i 165647 
15232 37186677 
,'5^33 37187507 
i5234 37188337 
J52353 7189167 



5236 3.7i§yyy6 
5237J 37190826 
5238.37191655 
5239!37192444 
5240 3 7193313 



52416.4194142 
i5242"a7194970 
S243j37195799 
5244 3 7196627 
5245J37197455 

■5246 
5247 
5248 
V5249 
5250 



o.i \\ fc:C83 
37199111 
3 7199938 
3 7200766 
37201593 



12 3 



LOGARITHMICK 



No-I Loer. 



52513.7202420 
5252*37203247 
5253*3 7204074 
5254 3.7204901) 
52553.7205727 ) 
5256 3.7206554. 
5257J3.7207380 
5258 1 3-72082061 

5259 3.7209032; 

5260 3.7209S5 7J 
526l'»3. 7210683 
52623.7211508 

5263 3.7212394 

5264 3.721U59 
5265,3 7213984 
5206,3.7214809 
5267|3.7215633 
5268 3.7216453 
5269:3.7217282 
5270 3.7218106 



No.| Log. 






5301 3.7243578 15351] 

5302 3.7244397||535: 
5303,3.7245216:5353! 

5304 3.7246035j 5354 

5305 3. 7246854 J 5355 

5306 3.7247672 15356 ! 
53073 7248491 5357J 
53083.7249309)5358 
5309 3.7250127 ! 5359' 
53103.7250945 5360 



5271,3.7218930 
527237219754 
5273 3.7220578 
5274 l 3.7221401 
5275 3.7222225 



53113.7251763 

5312 3.7252581 

5313 3.7253398 

5314 3.7254216 
53153 7255033 



3.7284350 
3 7285161 
3.7285972 
3.7286784 
7287595 
3 7288406 5406 



5361: 
5362 
5363 
5364 
53651 



Los- 



3 7289216 
3.7290027 
3.7290838 



No. 



5401 
5402 
5403 
5404 
5405 



3.7324743 
3.7325546 
3.7326350 
3.7327155 
3.7327957 



5407 
5408 
5409 



3.7291648.5410 



3.72924585411 
3.7293268 5412 
3.7294078 5413 
3.7294888 5414 
3 7295697i5415 



5316 3.7255850 

5317 3.7256667 
53183.7257483 

5319 3.7258300 

5320 3.7259116 



5321 3.7259933 5371 



5366 
53671 
5368' 
53691 
.5370, 



LO£. 



3.7328760 
3.7329564 
3.7330367 
3.7331170 
3.7331975 



3.7332775 
3.7333578 
37334380 
3.7335183 
3 7^35985 



No. Log. 



5451 
5452 
5453 
5454 
5455 



3.7296507J5416 3.7336787 
,3.7297316 5417 3.7337588 
3.72981255418 3.7333390 
372989345419 3.7339192 
37299743 5420 3.7339993 



5276 3.7223048 
527713.7223871 

5278 3.7224694 

5279 3.7225517 
52803.7226339] 



5322 3 7260749 

5323 3 7261565 
532413.7262380 
5325J3. 7263196 



5326 
5327 
5328 



53293.72.66457 



5330 



528113.7221162 
52823.72279841 
52833.7228806 
52843.7229628 
5285 3.7230450 



5286'3.7231272i 



5287 
5283 
5289 
5290 



*5372 
5373 
5374] 
5375 



3.7264012 
3.7264827 
3.7265642 



3.7267272 



5331 
5332 
5333 
5334 
5335 



3.7232093 
3.7232914', 
3.7233736 
37234557 



5336 
5337 
5338 
5339 
5340 



529113.7235378 

5292 3.7236198 

5293 3.7237019, 

5294 3.7237839" 

5295 3.7238660 



5296 
5297 
>5298 
5299 



3.7239480 
37240300 
3.7241120 
,3.7241939 



S.7300552',5421 3.7340794 



5376 
5377; 
5378 
5379 
5380 



5381 
'5382 
!5383 



3.7268087 
3.7268901 
3.7269916 
3.7270530 5384 
3.7271344 5385 
"»' 5386 



5341 
5342 
5343 
5344 
5345 



.7272158 
3.7272972 

.7273786 
3.7274599 
3.72 75413 
3.7276226 
3.7277039 
3.7277852 
3.7278664 
3 7279477 



3.7364762 
3.7365558 
3.7366355 
3,7367151 
3.7367948 



5456 
5457 
5458 
5459 
5460 



5461 

5462 
5463 



3.7368744 
3.7369540 
3.7370355 
3.7371152 
3.7371926 



3.7372722 
3.7373517 

13.7374312 



5464 C /375107 
5465(3.7375902 



3.73013605422 
3.7302168 5423 
3.7302977,5424 
3.73037855425 



3.7341595 
3.7342396 
3.7343197 
3.7343997 



3.7304593 5426 
3.73054005427 
3.7306208J5428 
3.7307015 5429 



3.7307823 



5300 '37242759 



5346 
5347 
5348 
5349 
5350 



3.7308630 
3.7309437 
3.7310244 
37311051 
3 7311857 



3.7312663 
37313470 
3.7314276 
3.7315082 
; 3.73 15888 



3.7280290 
3.7281102 
3.7281914 
3.7282726 
3.7283538 



5387 
5388 
5389' 
5390 

539l!i3.7316693 
5392] 3.7317499 
5393:3.7318304 

5394 3.7319109 

5395 3.7319914 



5430 
5431 
5432 
5433 
5434 
5435 



3.7344798 
3.7345598 
3.7346398 
3.7347198 
3.7347998! 



5466 3.7376696 
546737377491 

5468 3.7378285 

5469 3.7379079 
547037379873 



5471 

5472 
5473 
5474 
5475 



5477 



5479 
5480 



3.7348798: 
3.7349598 
37350397 
37351196 1 
3.7351995 



5436 
5437 
5438 
5439 
5440 



5441 

5442 
5443 
5444 



5396 
5397 
5398 
5399 
5400 



3.7320719 
3.7321524 
3.7322329 5448 
3.7323133 
3.7323938 



5481 

5482 
5483 



3.7380667 
37381861 
3.7382254 
3.7383048 
3. 7383841 
5476 3.7384634 



3.7385427 



54783.7386220 



3.7S87013 
3,7387806 



3 7352794 
3.7353593 
3.7354392 
37355191 
3.7355989 



3.7356787 
37S57585 
37358383 
3.7359181 



5445 3.7359979 



5446 
5447 



5449 
5450 



3.7360776 
3.7361574 
3.7362371 
3.7363168 
3.7363965 



3.7388598 
3.7389390 
3.7390182 
5484 3.7390974 
5485(3 7391766 



5486(3.7392558 

5487 3.7393350 

5488 3.7394141 

5489 3.7394932 

5490 3.7395723 



5491 
5492 
5493 
5494 
5495 



5496 



3.7396514 
3.7397305 
3.7398096 
3.7398887 
3.7399677 



3.7400467 



5497 3.7401257 
15498 3.7402047 
5499 3.7402837 
'5500'3.7403637 



ARITHMETICS 



229 



550613.7408302, ^37447622 



55073.7409151 
55083.7409939 
55093.7410728 
55103.7411516 
5511 £7412304 
551237413092 
5513 3 7413880 
55143.7414668 
55153.7415455 
55163.7416243 
55173.7417030 
55183.7417817! 
55193.7418604 1 
5520 3-741 9391 
5521 



Lor:. 



5557.3.7448404 
55583.74491 85 
5559.3.744990? 
55603.7450748 



556l'3.745l529 
5562 ! 37452310 
55633.7453091 
55643.7453871 |56143.7492724| 



5565 3 7454 652 ; 561 5j 3,7493498 



5566^7455432 J5616 .3.74^4271 



5567J37456212 



.74201771 

55223.7420964J 
55233.7421750 
55243.7422537 
55253.7423323 



55263.7424109 
552737424395 
55283 7425680 
552937426462 
55303.7427251 



5531J3.7428037 
5532)3.7428822 
55333,7429607 
55343.7430392 
55^537431176 
5536|3.7431961 
553713.7432745 
5538.3.7433530 



554OI3.7435098 



55433.7437449 



5539;.3.7434314 55893.7473341 



No., Log. 



£0, ^_ 

55013.7404416 555l'3.7443>12 
55023.7405206 5^523.7444495 
55033.7405995 55533.7445277 
55043.7406784 5554 ! 3.7446059 
550537407573 555537446841 



No. 



5601 
5602 
5603 



Log:. 



3.7482656 
3.7483431 
3.7484206 
56043.7484885 
560537485756j 
56063.7486531' 
5607I3.7487306 1 , 
560837488080 1 
5609374888541 



|5610 
5611 
15612 
'56131 



3.7489609 , 
3.7490403* 
3.7491177!, 
37491 950! 



5568'37456992 '5618 3 7495817 
5569,37457772 J5619 3.7496590 



557037458552 



5571,37459332 
5572^7460111 
5573 37460890 
55743.7461670 
5575 3.746244 9 
55763.7463228 
55773 7464006 
55783.7464785 



5621 3.749*lo6 

5622 3.7498908 

5623 37499681 

5624 3.7500453 

5625 3 750122 5 
5b2b 3.7501997 



5590 3 747411 815640 



joy; 
559: 



13 3.7476448 5643 



5544137438232 5594'3.7477225 J5644 



5547|S7440582 [5597 37479553 5647 37518178' '569? >. 
554813 7441365 (5598 37480329 '5648 3751 8947* '5698 3. 
5549 37442147 J5599 37481105 5649 37519716!J569C 3 



K55Q13 74429301)5600 3.7481880 5650 37520484' 57Q( 



56173.7495044 



5620 3.7497363 



{5639 



5541|3>435881 15591 3 7474895 J5641 3.7513561 5691 37551886 
5542 3.7436665 5592 37475672 ;5642 37514331, 5692 3.7552649 



5545 3.7439016 1 3595 3 7478001 ;5645 3.7516639 569513 7554937 ! 5745 
55463.7439799 J5596 3.7478777 '5646 O.7517409 569t 77555700 !5746 

37556462 [5747 
37557224 '57A& 
37557987 ;574S 
37558749 1 '5750 



1 Log, j 

0,7521253' 
3.7522022 
37522791 
3 7523558 
3 7524326 
37325094 
37525862 
37526629 
37527397 
3.7528164 

37529699 
3.7530466 
37531232 
3.7531999 
37532766 
37533532 
3.7534298 
; ; 7535065 
3.7535831 
37536596 
37537362 
. _ 37538128 
5674 3.753#893 
3.7539659 



po/oi 7540424 

5677 3.7541189 

5678 3.7541954 
'56793.7542719 
15680 3.754348 3 
J5681 3.7544248 



5721 
5722 
5723 
5724 
5725 



5627 3.7502769 
562837503541 

5579 3.7465564 15629 3.7504312 

5580 37466342 5630 3.7505084 
5581*3.7467120 5631 3.7505855 

5582 3.7467898 5632 3.7506626 

5583 3.7468676 563 

5584 3.7469454 5634 37408168 ,f 5684|3 7546541 

5585 3.7470232 |5635 3 750893 9^685 3-7547305 

5586 57471009 '5B36 3.7509710 0086 J.7 o4aoo9 

5587 3.7471787 5637 37510480 15687 3.7548832 

5588 3 7472564 5638 37511251: 5688 3 7549596 



^5682 37545012 

3 7507398|'5683 3 7545777 



37512021 J568< 



3.7512791.5690 3.7551123 



3.7515101 H5693 
37515370 5694 



3 7550359 



3.7553412 5743 
3 7554175 |5744 



No. 
5701 
5702 
5703 
5704 
5705 



Lop\ 

3.7559510 
37560272 
3 7561034 
37561795 
37562 556 
3 7563318 
3.7564079 



5709 



15711 
! 5712 

J5713 
'5714 
;5715 



5/0(5 
5707 

570837564840 
37565600 



57103.7566361 



3.7567122 
3./*67882 
3.7558642 
37569402 
7570162 



;5716jj.f 570922 
.571737571683 
'5718:3.7572442 
571937573201 
;5720|3 2573960 



5726 
572 
5728 
'5729 



o., 5/4719 
37575479 
17576237 
37576996 
. 757 7755 
J.7573513 



5732 
5733 
'5734 



37579272 
3.7580030 
3.7580788 

573013.7581546 

'5731 



o./ 5*2304 
37583062 
37583819 
37584577 



573337585334 



j/J6 
, : 5737 
J5738 
<5739 
5/40 



^/"3boUyl 
3758684f 
3.7587605 
3.7588362 
58911? 



i574i 



5 l 3&yti75 



J742 37590632 



37591388 
37592144 
3.7592900 
3.7593650 
37594412 
3.7594168 
1.7595923 
3.759 667? 



30 



I 



2St 



L0GAR1THMICK 



575137597434! 58U1 



5752 3 7598189 ;5802 3 76C>5777\<5852 3 7673043 



57611-3.76049/9 
5762*7605733 



5763 



3 7606486 5813 3.7644003 



57643.7607240 

57653.760799: 

5766 

5767 

5768 

5769 



3.7608746 
37609500 
3.7610253 
3.7611005 
5770 3.7611758 
3.7612511 



5771 
5772 
5773 



3.7613265 

3.7614016/5823 

57743.7614768 
577537615520 
57763.7616272 
57773.7617024 
^778 37617775 
577937618527 
578037619288 



5781 
5782 
5783 
5784 
5785 
5786 
5787 



5789 
5790 
5791 
5792 
5793 
5794 
5795 
5796 
5797 
5798 
5799 
5800 



5814 
'5815 



37620030 
37620781 
37621532 
37622283 
37623034 
37623784 
37624535 



578837625285 



37626055 
7626786 



5804 
5805 



5753 3 7598044!j580: 
5754!37599699r 
5755:37600453 
57563.7601208 
57573760196$ 
57583 7603714 
575913 7603471 
5760'3 7604225 



No, Lutf. ||No[ Lo?. ||i\o.| L,ug. 



37635U29|(5851 37672301 



3.7636526 J5853 37673785 
3 7637274 f 5854 37674527' 
3 7638022 15855 3 7675W9 



5807 
5808 
5809 
5810 
5811 
5812 



5901 
5902 
5903 
5904 



58(58 £7638? /7poo6 376/6011, 
-' 3.V 6395181 5857,37676752 



37640266 
37641014 
37641761 



3.7642509 
37643256 



3.7644750 
37645497 



5816 
5817 
5818 
5819 
15820 



37646244 
37646991 
37647737 
37648484 
3 7649230 
,5821 37649976 
;5822(37650722 

3.7651468 

582437652214 



58253765295S 



582637653705 
1582737654450 
58283.7655195 
'582937655941 
583Q 3 765668 6 
5H3i : 37657430 
5832.37658175 
583S37658920 
{583437659664 
5835 ! 3 7660409 



I5d3g 3 766115 

,5837.37661897 

583837662641 

583937663385 

584037664128 



37627536 5841J3.7664872 
37628280 58423766561f 
1.7629035 58433 7666351 
37629785} 584437667102 
3 7630534; 5845 37667845 
584(>37668588 



37631284 ! 
3.6632033 
37632782 
37633531 
3.7634280 



5847,3.7669331 
5848,37670074 
5849.37670816 
5850 ! 3 7671S59 



5858'37677494] 
585937678235^ 



5860 



376789761 



566i 3767971/1 



5862 
5863 
5864 
5865 



37680458J 
37681199 
37681940 
37682680 



5866 
5867 
5868 
5869 



37683421 
3.7684161 
37684901 
37685641 



5870 37686381 



5871 
5872 
5873 
5874 
5875 



Log. [liNo. ^Lug. 



37709256: 5951 3.7745899 \ 
37709992! (5952 37746629 
37710728: 5953 3.7747359 
37711463; 5954 37748088 



5905 , 37712199 , !5955 

5906 ! 3 7712934, joy56 

5957 

5958 

5959 



59073.7713670 1 



5908 , 37714405 l 
5909;3.7715l40i 
5910 3 7715875, 
5911 



5912 
5913 



377166101 



o.774^547 
37750276 
37751005 
37751734 
5960 37752463 



37717344 
37718079; 
5914J37718813 
J5915 3.7719547 
1591637720282 
[591737721016 
5918 , 37721750 
59193.7 722483 
5920 i 3.7723218 



376871211 
37687860 
37688600 
37689339! 
37690079! 



5tt/ 6 3769081 81 

58773.7691557 

587837692296! 

587937693035 ; 

588037693773' 

5881 

5882 



37694512 
37695250 
588513769598a 
5884)37696727 
5885137697465 
58*5(^37698203 
5887(37698940 
5888i3.769667fr 
5889J3760O415 
58903 7701153 ' 
589l'377018<*0i 
5892137702627 
58933.7703364 
58943.7704101j 
5895377048381 
589637705575 1 
589737706311 
58983 7707048 
589937707784; 
59003.7708520! 



5927 
5928 
5929 
5930 
5931 
5932 
5933 



592137723951 



7724684 
37725417 

5924 3.7726150 

5925 37726183 



5922 
5923 



592613.7727610 



37728349 
37729082 
37729814 
37730547 
37731279' 
37732011 
37432743 
593437733475 
5935 b 7734207 ] 
593613 7734939; 
5937J3 773567 



593937737133; 
594037737864 



5941'37738595 
59423.7739326 
594337740057 
594437740?88 
59453.7741519 



5973 
5974 



37748818 



5961 

596$ 

5963 

5964 

5965 

5966 

5967 

5968 

5969 

5970 

5971 37760471 

59723 7761198 
3.7761925 
37762653 



37753191 
3 7753920 
37764684 
3.7755379 
3 7756104 
3.7756832 
37757560 
37758288 
37759016 
37759743 



59753 7763379* 



5976 
5977 
5978 



3776410^ 
3.7764833 
3J7G5559 
597937766285 
5980 i 3.7 767012 



5^1:3.7767738 
5982J37768464 
5983:3.776919(1 
5984'37769916 
5985J 377706 41 
59863.7771367 
5987J37772093 



5938 37736402) 5988,37772818 



598937773543 
5990' 3777426 8 
5991,3.7774993 
5992!37775718 
5993!3777644S 
59943.7777167 
599537777892 



594637742249 
5947 37742979 
594837743710 
94937744440 
5950 3774517Cfl6000!3 7781512 



59963.7778616 
5997137779340 
599837780065 
599937780781 



2l 



ARITHMETICS 



331 



6001 
6002 
6003 
6004 
6005 



6006 
6007 
6008 
6009 

6010 

6011 

6012 

601 

6014 

6015 



0016137793078 
601713 7793800 
60183.7794522 
60193.7795243 
6020 ! 37795965 



jNo. 



Log. 



i.7782236 
3.7782960 
3.7783683 
3. 7734407 
37 785130 

3-7785853 
3.7786576 
3.7787299 
3.7788022 
3.778874? 



6106 

3.7822576 6107 37838279 

37823293 6108 , 37858990 

3.7834009 6 ! 09137859701 

60603.7824726 I 6110!3.7860412 



37789467 
3.7790190 
37790912J 
' 7791634 1 ; 
3.7792356 f 



6061 3.7825443| 011113.7861 123 

6062 3.7826159p6il2l3.7861833 

6063 3.7826876 S 11313.7862544 

6064 3.7827592 '61 14 3 7863254 
6O65I3 7828308 1611 513.7863965 



6066 
6067 
6068 
6069 
6070 



OJ^lj3.7796686 

6022 3.7797407, 

6023 3.7798129 
60243.7798850 
60253.7799570 



'60263.7800291 
6027;3.7801012 
6028'37801732 
602913.7802453 
6030'3.7803173 



60313.7803893 
60323.7804613 
5033*3.7805333 
60343.7806053 
6035 3.7806773 



60363.7807492 
6037J3.7808212 
60383.7808931 
60393.7809670 
6040'37810369 



60413.7811088 
60423.78118Q7. 
6043i3.7812526' 
6044 , 37813245 
6045 3.7813963 



60463.7814631' 
6047,3.7815400 
60483 7816118 
604937816836 
60503.7817554 



No. 



0051 
6052 
6053 
6054 
6055 



3.7818271 
3 7818989 
37819707 
3.7820424 
3.7821141 



6056 
6057 
6058 
6059 



3 7821859 



Lt>£. 



£ 

v 



No.j Log. 



61013.7854010 
161023.7854722 
6103 3.7855434 
61043.7856145 



6105 



3.7856857 



378o756b 



No.] Ley. 



[No. 1 Lo£. t 



16151 3 7889457 
|6152|3 7890163 
161533.7890869 
6154;37891575 
6155'S.7892281 



37829024 
3.7829740, 
3.7830459' 
37831171 
3 7831887! 



1 

16156,3 7892986 
6157 3 7893691 
615837894397 
6159:3.7895102 
616037895807 
^161,0.7696512 
61623 7897217 
,6163'3.7897922 
|6164;3.7898628 
61653.7899331 



16201 $.,924617. 
.62023.7925318 
'6203 3.7926018 
162043.7926718 
£205 3 7927418 



6206 3.7 9-^8118 
62073.7928817 
'62083.7929517 
62093.7930217 
62103.7930966 



6071 
6072 
6073 
6074 
6075 



6076 



0116 
6117 
6118 
6119 



„„* 37866804 

6120 3 7867514 



3.7832602 
S7833318 
3.7834033 
3.7834748 
.7835463 



o.? 004075 
3.7865384 



:61 663.7 900035 
16167 



3.7866095 ;6168 ! 3.7901444 



■6J69 3.7902148 
6170,3.7902852 



6121 f 3.7808224 
6122J3 7868933 
6123 3.7869643 
6124'3.7870352 
6125(3.7871061 



1.7836178 



60773.7836892 
60783.7837607 
60793.7838322 



6080 37839036 (6130J37874605 
6081 



37839750 



6171137903555 
6172J37904259 
6173 ! 37904963 
6174 , 37905666 
6175;37§06370 



6126137871770 
6127 3.7872479 
612837873188 
6129137873896 



6131:3.7875313 



608237840464 6132'37876021 
6083 3 7841178 f 6133;3.7876730 
6084137841892 61343 7877438 
6085 37842607 16135J3 7878846 



608637843319 
6087}37844034 
6088'37844746 
6089J3 7845460 
60901 37846173 
6691|37846886 
609237847599 
60933.7848312 
609437849024 
60953 784973^ 



613637878853 
6137J3.7879561 
613837880269 
613937880976 
61403 7881684 

6141 37882391 
614237883098 
61433 7883805 



6096 
6097 



6099 



61913.7917608 

6198*37918309 

6193 3 7919011 

61443 7884512^619437919712 



37850450 
37851162 



609837851874 



37852586 



1610037853298 



6145 3788521^ 



614637885926 
6147;37886632 
614837887339 
61493.7888045 
6150 ; 37888751 



0176J3.7907073 
161773.7907776 
61783.7908479 
6179!3J'909182 
6180J37909885 



618137910587 
61823 7911290 
618637911993 
618437912695 
6185 37913397 



618637915099 
61873 7914801 
61883.7915503 
618937916205 
6190 3 7916906 



6195 37920413 



6196o.-9211l4 
619713.7921816 
6198|'j.792251f 
519913 792321F 
620013.7923917 



0^113.7931615 

6212 37932314 

6213 37933013 
621437933712 
6215 3.7934411 



62l6o./y35*l0 
6217J3 7935809 
6218 , 3.7936507 
6219:3.7937206 
6220'3 7937904 



0221 37*38602 
622237939300 
6223'3.7939998 
J 6224 ! 37940696 
'6225:37941394 



6226*37941091 
6227*37942789 
»6228 ! 37943486 
I6229J37944183 
i6230!37944880 



;6231 
'6232 
|6233 
6234 
6235 



37945577 
37946274 
37946971 
37947668 
37948365 



0236j3.7i/49061 
6237 3 7949757 
|62383 7950454 
763393.7951150 

6240!3,7P51«46l 



,6241(3.7^52542 
6242'37953238 
! 6243;3 7953933 
6244 l 37954629 
! 6245 l 3 7955324 



6246 3./ 956020 
6247*37956715 
624837957410 
624937958105 
6250 3 7958800 



232 



LOGARITHMICK 



No.l Lop:. 



No. | Lot: 



6252 
6253 



62543 7961579 



6255 



6256|3 
62573 
6258k 
6259!3 
62503 



626113 
6262!3 
6263'3 
62643 
6265'3 



62*63 
62673 
62683 
6269*3 
6270'3 



3 7960190 
3 7960684 



\'o.( Lor. 



|No. Log. 



3 7962273 



7y629b, 

7963662 

7964356 

7965040 

7965743 



79o<5437 
7967131 
7967824 



66'jl.i /u9WJ5- 

6302 ; 3 799478416352 
63033 7995473 16353 3 
63043 799616211635413 
;63053 7996rt51; 1 6.iJ5 



163513 80284^1 1 
8029105 

802978'J ; , 
8030472; 
8031155>|6405 



64U13 8062478 
6402 ! 3 806315? 
640£ 



6306,3 
5307,3 

6som 

J6309J3 
631 ^3 



3 8063835 

6404 3 8064513 

8065191 



?y9/D4uji63563 
7998288' 6357 3 
79989171:6358 3 
7999605] .6359 3 
8000 v ")4i 636013 



,'6jiiu3 
631213 
631313 



80318J9J 
8032522. 
8033205! 
803:^888! 
80J4571 



6406 
6407 
6408 



7968517|i63143 
7 6 ( >2U , 6315|3 



800096^|j63Gl 
8001670 6362 
8002358 Ji 6363 
8003046j:63643 
8:)03r.°>4! ! .6365!3 



8035264 
8035937 
8036619 
8037302 
8037984' 



6J713 
62723 
6273*3 
6274'3 
6275-3 



7963*04 

7970597 

7971290 

7971983 

7972575 

7973368 

7974060 j 

7974753 

7975445 

7976137 



f 6316i3 
6317|3 
163183 
163193 
"63203 



6270 3 
627713 
62783 
6279;3 
62803 



7976829|! 

7977521 

7978213 

7978905 

79795961 



6281 ; 3 
6282,3 
62833 
6284'3 

6285 3 



7980281 
7980979 
7681671 
7982362 
7983053 



62863 
628713 
6288!3 
62893 
6290 3 



7983744 
7984434 
7985125 
7985816 
7936506 



6291 3 
62923 
62933 
92943 
62953 



7»67 M 
7987887 
7988577 
798926? 
7088957 



62963 7990647 
6397j3 7991337 
62983 799202? 
62993 7992716 
6300'3 79934051 



8U04421 
8005100J 
8005796 
8006484 
8007171 



636o!3 
63673 
63683 



6369! 
6370J3 



6321 
'6322 
16323 
16324 

j-5325 



8U07858 
8008545 
8009232 
8009919} 
8010605! 



8038666, 
80393481 
8040030' 
8040712 
8041394 



63263 8011292 
8011978 
8012665* 
8013351 
%014037 



|6327 
6328 
6329 
6330 



5(63313 
,63323 
163333 
63343 
6335 3 



i6336 
63373 
6338 
6339 

6340 



6341 
6342 
6343 
6344 
6345 



63463 
6347 
6348 
6349 



63503 



8UU723 
8015409 
8016095; 
8016781! 
8017466 



8018152 
8018837! 
8019522 
8020208' 
8020893 



6371J3 
63723 
6373)3 
63743 
6375\3 



8042076| 
8042758: 
8043439 
8044121 
8044802 



6381 
6382 
6383 
6384J3 

6385 



63863 
63873 
63883 
63893 
63903 



8021576! 
80£2262 : 
80229471 
8023032! 



6391 



63923 

6393 

6394 



3 8024316-6395 



8025001 
8025685 
8026369 
8027053 
8027737' 



6396 3 

6397 3 
63983 

6399 3 

6400 3 



\o.| Lo£. 



3 80658o9 
3 8066547 
3 8067225 
6409:3 8067903 
641013 8068580 



6411J3 
64123 
641313 
641 4'3 
6415 3 



64163 
6417 3 
64183 
641913 
6420*5 



8069258 
8069935 
8070613 
8071290 
8071967 



6451 



f6452 
6453 



6461}3 
6462 ! 3 
6463 ! 3 
6464!3 
646513 



8072644 
8073320 
8073997 
8074674 
8075350! 



64213 
642213 
64233 
64243 
64253 



8076027) 
9076703- 
8077379 
8078055 
8078731 



6376'3 8045483] 6426'3 8079407 
63773 8046164 64273 8080083 
63783 8046845164283 8080759 
63793 8047526 rt «4293 8081434 
63803 8048207 6430}3 8082110 



8048887 
8049568 
8050248 
8050929 
8051609 



8052289 
8052969 
8053649 
8054329 
8055009 



8055688 
8056368 
8057047 
8357726 
8058405, 



6431|3 
64323 
6433*3 
64343 
643513 



8082785' 
8083460 
8084135 
8084811 
8085485 



643613 
643713 
6438'3 
6439.3 
6440 ! 3 



8086160; 
8086835 1 
8087510! 
8088184 



8059085] 
80597631 
8060442 
8061121 
8061800 



644113 
6442-3 
6443J3 
644413 
6445 J3 
6446.3" 
644713 
64483 
6149,3 
6450' 3 



8089533; 
8090207 
80908811 
8091555: 
8092229| 



6496 3 



8092903,' 
8093350 
8094250 
8094924; 
S095597'65003 



3 8096270 
3 8096944 
3 8097617 
6454'3 8098290 
6455»3 8098962 



64563 8099635 
6457 ! 3 8000318 
64583 8100980 
6459'S 8101653 
6460*3 8102325 



8102997 i 

8103669 

8104342 

8105013 

8105685 



6*6613 
6467|3 
64683 
64693 
64703 



8106357 
8107029 
81077tX) 
8108371 
8109043 



647i;3 
6472J3 
647313 
6474'3 
6475 3 



6476 



3 8113068 
8113739 
64783 8114409 

3 8115080 
64803 8115750 



64773 



6481 
6482 
6483 
6484 
64853 



64863 

6487. 

64883 

6489 

64903 



6497 3 
64983 
6499 



8109714 
8110385 
8111056 
8111727 
8112398 



8116420 
8117090 
8117760 
8118430 
8119100 



1*119759 
8120439 
8121108 
8121778 
8132447 



64913 8123116 
64923 8123785 

6493 3 8124454 

6494 3 8125123 

6495 3 8125792 



8126460 
8127129 
8127797 
8128465 
8129134, 



ARITHMETICS. 



23* 



6501 



65023. 



3.8129802! 
8130470! 
650313*8131137 
65043.8131805 
6505 3.8132473 



6506 
6507 



3.8133141 
3.8133808 
6508T3.8134475 
6509|3.8135143 
651013.8135810 



65113.81^0477 
65123,8137144 
6513 3.8137811 
65143.8138477 
65153.8139144. 



No 



Log. ( 



6556 3.8166389 
3.8167052 
3.8167714 
3.8168376 

65603.8169038 



6557 
6558 
6559 



6501 



6562 
6563 



651$3.8l39dll| 
6517;3.8140478 
65183.8141144 
6519-3.8141810 
65203.8142476 



6521^3.814^14^ 
65223.8143808, 
6523;3.8144474 i 
65243.8145139i 
65253.814580? 



65263 8146471 
6527|3.8147136 
6528^.8147801 
652913.8148467 
6530;3 8149132 



65313 8149797 
65323.8150462 
6533 3.8151127 
65343.8151791 
65353.8152456 



6536 
6537 
6538 
6539 

6540 



3.815312U) 
38153785 
3.8154449 
38155113 
'138155777 



6541 
6542 
6543 
6544 



654b 
6547 
6548 



3.8156441 
3.8157105 
3.8157769 
3.8158433 



6545 3.8159096 



3.8159760 
38160423 
38161087 

6549^3 8161750 

6550 3 



No. 



6551 

6552 

6553 

65543. 

6555 



Lo£. 



3.8163076 
3.8163739 
3.8164402 
1.8165064 
3.8165727 



3.8169700 



3.8170362 
3.8171024 
65643.8171686 
6565J3.8172347 



6566-3 8173009 
6567)3.8173670 
6563-3.8174331 
6569|3.8174993 
657038175654 



6571^38176315 
6572J3.8176975 
6573!>8177636 
6574:3 8178297 
6575.3 5178958 



No. 



6601 3.8196097 

6602 3.8196775 

6603 3.8197413 

6604 3.8198071 

6605 3.8198728 



6606 
6607 



3.8199380 
3.8200043 
66083.8200700 
66093.8201357 
66103.8202015 



6611 
6612 
6613 



6576J3 81796181 
657*3.8180278! 
657a : 3.8180939! 
6579|3.8181599 
65803.8182259! 



6581 3.8182919J 
65823.81835791 
6583 3.81842391 
65843-8184898 
6585J3.8185558 



5861; 



658613.8186217 
6587i3.8136877 
6588!3.8187536 
65893.818S195 
65903.8188354 



659a 38189513; 
659VJ3.8190172I 
5593 3.8190831 
65943.8191489 
6595 3.8192148 



6596,3.8192806 
6597 3*8193465 
6598.3.8194123 
6599 ^.8194781 
8162413W6600 3. 8195439 



Lo*. 



3.3202672 
3.8203328 
3.8203985 
66143.8204642 
66153.8205298 



66163.8205955 
66173.8206611 
66183.8207268 
6619-3.8207924 
6620'3.8208530 



6621 3.820 l J236 

6622 38209892 

6623 3.6210548 
3.8211203 
38211859 



6624 
6625 



6626 
6627 
6628 
6629 
6630 



6631 
663! 
663i 
663' 
6635 



6637: 
6638 
6639 
6640 



6642 
6643 
6644 
6645 



6646 
6647 
6648 
6649 
6650 



3.8212514 
3.8213170 
3.8213825 
3.8214480 
38215135 



3.8215790 

6632|a8216445 

.8217100 

6634)3.8217755 

3.8218409 



66363.8219064 
3.8219718 
5 8220372 
58221027 
5.8221681 



6641 3.8222335 
3.8222989 
3.8223642 
3 8224296 
3.8224950 



3.8225603 
.8226257 
3.8226910 
3.8227563 
3.8228216 



I No. * Log. | 



6651 13, 
66523. 
6653'3. 
6654J3- 
6655\3. 



,8228869 i 
8229522 
8230175! 
8230828' 
8231481' 



670113.8261396 
6702J3.8262044 
6703|3.8262692 
6704*3.82633401 
6705'38263988 



6656 ! 3 
665713. 
66583, 
66593. 
6660 3. 



8232133! 
,8232786. 
,8233438 
8234090 : 
3234742' 



66613.8235394 
6662 3 8236046 
666338236698 
66643.8237350 
66653 8238002 



66663.8238653 
66673.8239305 
66683.8239956 
66693.8240607 
66703.8241253 



6671 3 8241% -J 
'6672 : 3.8242560 
'66743.8243211 
6674*3.8243862 
6675 3 8244513 



6676 3.8245163 
,66773.8245814 
66783 3246464 
16679 3.8247114 
; 66S0 3.8247765 



6681 3.8248415 
66823 8249065 
66833.824971 
66843.8250364' 
6685 3.8251014: 



668638251664' 
6687 3.825231 
66883.8252963 
66893.8253612; 
66903.8254261 s 



66913.8254910 
6692i3.8255559 t 



6693 
6694 
6695 



6696 
6697 
6698 
6699 
6700 



3.8256208J 
3 8256857! 
3 8257506 



3.&258154! 
3.3258803! 
3.8259451i 
3.82601001 
3 8260748' 



No*| Lor. 



6706 j 0264635 
6707,3 3265283 
6708 3 8265931 
6709J3.8266578 
671013.8267225 



6711 
6712 
6713 
6714 



3.8267873 
3.8268519 
3.8269166 
3.8269813 



6715 38270460 



6ri6i3.827ll07 
6717|3.8271753 
6718 ! 3.8272400 
6719)3 8273046 

6r203.t>~"3693 



67 2113.82? 4339 



6722 
6723 



£8274985 
3.8275631 
672413.8276277 
6725*3 8276923 



6726J3 8277569 
67273.8278214 
3.8278860 
3.8279505 



6728 
6729 



6730 3.8280951 



6731 
6732 
6733 

6734 
6735 



,8280769 
3.8281441 

8282086 
3.8282731 
38283736 



6736J3.8284021 
6737|3.8284665 
6738J3.8285310 
673913 8285955 
6740'[3 8286599 



67411J.6237243 
6742;3 8287887 
6743 ( 3.8288532 
6744i3.8289176 
6745 J 3 K2898 20 
6746 o b^ii0463 
674713 8291107 
6748 ! 3 8291751 
6749'3.8292394 
6750 3 8203830 



334 



LOGARITHMIC* 



6751 3 82*i(x>i j68Ul 
67523.8294324 6802 
67533 4294967,6803 

6754 3.8295611 I |6804 

6755 3J296254 
67oo 3 8jy6oyo 

3.8297539 
3.8298182 
3S298824 



6760!3 8299467 



676l!3.83lK>iu* 
676213.4300752 
6763J3.8301394 
67643*8302036 
6765!3.8302678 



No.f Loe:. .No. 



3.8325728 
3.8326366 
38327005 
3.8327643 
69053.8328281 



6807 
6808 
6809 



6813 



68153 8334659)6865 



670oj3 8303320 
6767 3-8303962 
67683.8304604 
67693.3305245 
6770' 3.83 5887 
6771|3.83J6o^o 
67723.8307169 
67733.8307811 
6774 , 3-8308452 
6775(3.8309093 



68l0j3.8335296 
6817|3.8335933 
68183.8336570 
681938337207 
6820 ! 3.8337844 



67763.8309/^4 
67773.8310375 
67783.8311016 
67793.8311656 
67803.8312297 



6781]3.83129o7 
6782 ! 3.8313578 
6783'3.8S14218 
67843.8314858 
6785 3.8315499 



I.o^. 



6o\>0 3.8328yiy 
3 832*558 
3 8330195 
3.8330833 
6810 3 8331471 
6811 3.833^oy 



2.8333384 



68143 8334021 



No., Love- 



68 J 1,3.8357540 
6852:3^358174' 
6853 3.8358607J 
68543.8359441 1 
6855 3.8360075' 



69013.838 120 
69023.8389750 
69033.8390379. 
69043.8391008 
6905 3.8391637 



68o6 3.8360708 
6857|3 8361341! 
6858 3 8361975i 
685913.8362608 
6860'3835324l! 



6861 



3.8^63874 



6812 3 8532746*6862 3 8364507! 



16863 3.8365140' 



6864 



38365773: 
3 8366405 



6866 3.8367038 



6867 
'6868 
6869 



8367670. 
3.8368303 
3.8368935^ 



6870 38369567; 



6821 , 3.83384riU 
68223 8339117 
68233.8339754 
68243 8340390 
6825 3.8341027 
68263.8341663 
68273.8342299 
68283.8842935 
68293.8343571 
68303.8344207 



6871 
16872 
J6873 
6874 
16875 



3.8370199; 
3.8370832; 
3.8371463 1 



3.8372095 6924'3.840354ll 6974 



m?6 

6877 



6879 



68313 8344443 
6832 3.8345479 
683338346114 
6834*3.8346750 
6835,38347385 



6786 3.831613* 16836,3.8348021 

6787 3.8316778 i6837;3.8 348656 

6788 3 8317418 I683&3.8349291 

6789 3 8318058 1683938349926 

6790 3.8318698 168403.8350561 



6/yi 3.83iyoo/i 
67923.8319977 
67933 8320616 
67943 8321255 
67953.3321895 



6841'3.8351iyo 
6842:38351831 
6843'3 8352465 
6844 ] 3.8353100 
68453.8353735 



67963.832268-* 
679738323173 
67983.8323812 
67993 8324450 
6800 3.8325089 



6882 
6883 



6885 



6886 
6887 
6888 



69li 3.8395409. 
69123.83960371 
69133.8396666! 
69143.8397294' 
6915 3 8397922 



3.8372727, 



3373359; 
3.8373990 



68783.8378622' 



3.8375253: 



68803.8375884 



6881 3.8376510 



3.8377147 
3.8377778 



68843.8378409 



3.8379039 



3.8379670 
3.8380301 
3.8380931 



6889 3-8381562 



6890 



3-8382192 



6891 
,6892 



.8382822 
3 8383453 



68933.8384083 



6894 
6895 



68463.835436y 
684713 8355003 
6848'3 8355638 
684913.8356272 
6850 , 3.8356906 



No. f Logi 



No. 



6951 
6952 
6953 
6954 
6955 



69063 83*2266 
69073.8392895 
6908*3.839352^ 

6909 3.8394152' 

6910 3.8394780' 



69o6 
6957i 
6958 
6959 
6960 



6961 
69621 
6963 1 
6964 
6965! 



6916 3.8398550 
6917S.8399178 
69183.8399806 
6919:3.8400453 
69203.8401061' 



692i;3.8401688 
69223.8402316' 
6923 8.8402943 1 



6925'3. 8404198 » 
6y2b;3~8404825 



6927 
6928 
2929 
6930 



3.8405452 
38406079, 
3 8406706 1 
3 840733 2] 

3.840795yi 



,6933 
! 6934 
6935 



6931 
169323.8408586 



3.8409212 
3.8410988 
3.8410465 



3842047, 
3 £421098 
3.8421722 
38422347 
3.8422971 



3.8423596 
38424220 
3.8424844 
3.8425468 
3 8426092 



6975 



6976 
6977 
6978 
6979 
6980 



Log. 



3.8426716 
38427340 
3.8427964 
3.8433588 
3.8429 211 
38429835 
3.8430458 
3.8431081 
3.8431705 
38432328 



6981 
6932 
6983 
69841 
6985 



3.8432951 
3.843S574 
3.8434197 
3.8434819 
3843544 2 
3.S436061 
.8436687 
3.8437310 
3.8437932 
3.8438554 



3.8439176 
3.8439703 
3.8*10420 
3.8441042 
3.844166> 



6936 3.8411091jj6986i 
16937 3.8411717l|6987i 

6938 3.841234316988 

6939 3 8412969! 69891 
6940;33413595j!6990' 



3.8384713 
3 8385343 ; 
8385973; 
3.8S86602 
3 838 232, 
38387861 



6896 
6897 
6898 
6899 
69003.8388491 



694113 8414220J6991, 
6942J3 84t4846p6992 
694^3 ?415472| ! ,6993 
694438416097 6994 
6945 3^416723 6995 



3.8442286 
38442907 
3.8443529 
38444150 
3.844477 2 
3.8445393 
3.8446014 
38446635 
3 8447256 
3.8447877 

3.8448498 
3.8449119 



69463-8417348 6996 
69473'8417973||6997 

6948 3-841859816998 '3.8449739 

6949 3-8419224 ,6999 i3.8450360 

6950 S.a±19848! l 7000 t3.843Q98Q 



ARITHMETICS 



3» 



No.f Log* , 

700l3lJ45llKiI 
0023.8452221 
70033.8452841 
70043 8453461 
70053.8454081 



70063.8454701 
70073.8455321 
70083.8455941 
70093.8456561, 
701034457180 



70113 8457800 
70123.8458419 
70133.8459038 
70143.8459658! 
7015|3.8460277 



No.. Log. No.| Lour. 



70513-8482507 
70523.8483123 
70533.8483739 
70543.8484355 
7055 ] 3 8484970 



7056 
7057 



71013.851319^ 
71023.8513807 
71033.8514418 
71043.8515030 
71053.8515641 



3.3485586 
3.848620r 
70583 8486817 
70593.8487432 
70603.8488047, 



70163.8460896 
70173.8461515 
70183.8462134 
70193.8462752 
70203 8463371 



70213 8463990 
70223.8464608 
3.8465227 
3.8465845 
3.8466463 



7023 
7024 
7025 



7036(3.8467081 
70273.8467700 
70283.8468318 
3.8468935 
3 8469553 



7029 
7030 



7033 
7034 



7041 
7042 
7043 
7044 
7045 



7046 
7047 



70313.8470171 
70323 8470789 
38471406 
3.8472024 
7035*3.8472641 



70363.8473258 
7037 3.8473876 
70383.8474493 
7039 3.8475110 
70403 8475727 



3.8488662 
3.8489277 
3.8489892 
70643.8490507 
70653.8491122 



7061 
7062 
7063 



7067 
7068 



7071 3.8494808 



7072 
7073 
7074 
7075 



6077 
7078 
7079 
7080 



7106-3.851625. 

71073.8516863 

71083.8517474 

710938518085 

71103.8518696 



71113.8519307 
71123.8519917 
7113 3 8520528 
71143.8521139' 
7115 3 8521749 



7066)3.8491736 
3.8492351 
3.8492965 
70693.8493580 
70703.8494194 



71163.8522359 
71173.852*970; 
71183.8523580 
71193.8524190! 
712038524800 



71213. 
3.849542*71223. 
71233. 
71243. 
71253. 



3.8496037 
3.8496651 
3.8497264 



7076)3.8497878 

.8498492 

r.8499106 

3,8499719 

8500333 



7081l38500946 



7082 
7083 



7085 



3.8501559 
3.8502172 



7084,3.8502786 



3.8503399 



7086:3,8504011 
7087^3.8504625 
70883.8505237 
7089'3.8505850 



3.8476343J 70913.350707 5. 
3 8476900 7092,3. 8507687; 
3.8477577! 7093:3.8508300 



3.8478193 
3 8478810 



709413.8508912 
70953:8509524 



No. 1 Lor. 



7151,8.8543668 
71523.8544275 
i7153'3.8544882 
715438545489 
71553.8546096' 



7156U8546703 
8.8547310 
3.8547917 
3.8548524 
3 8549130 



7157 
7158 
7159 
7160 



8525410 
,8526020 
8526629) 
8527239: 
8527849, 



71263. 
71273 
71283 
712*3. 
71303 



,8528458 
8529068! 
8529677 
.8530286 
8530895 



7161 
7162 
7163 
7164 
7165 



3.8549737' 
3.8550343 
3.8550950 
3 8551556 
3.8552162- 



7166 



7168 
3169 
7170 



7171 
7172 
7173 
7174 
7175 



7176 
7177 



3.8552768; 



71673.855; 



374 
3.8553980 1 
3.8554586 
3 85551921 



3.85557971 
3.8556403 
3.8557008: 
3.8557614 
3 8558219 



8.8558824; 
3.8559429J 



71S1 3.8531504 
7132 $.8532113 
71333.8532722 
31343.8533331 
713538533940 



71363.8534548 
7137J3 8535157 
71383.8535765 
7139|S.85S6374 
7090j3.8#)S462 ]7140,3 8536982 



7141 
7142 
7143 



3 8479426 70963 8510136 
3.8480043J 7097J3 8510748 
7048*3.8480659 70983.8511560' 
7049 3.8481275 7099J3.8511972 
705013 8481891 1 7100,3.8512583 



|71463 

1714; - 

(71 18)3 
7149 



3,8537590 
3 8538198 
3 8*38807: 
714438539414 
7145 3 8540022! 



8540630 
38541238. 

8541845 
3.8542453 

8543060 



7178.3.8560035 
7179,38560640; 
7180 3 8561244: 



17181 3 85*1849 
71823.8562454 
7183 3.8563059 
718438563663 
7185 3.8564268 



7186 l 3.8564872 
718713.8565476 
71883.8566081 
7189J3.8566685 
7190 3 8567289 



7191 3 8567893 
7192 3.856784? 
7193'3.8569101 
71943,8569704 
7195,3.8570308 



Nof L 



£• 



7201io.a573i,28 
7202o.857453I 
72U3 3.8575134 
7204 3 8575737 
7205'3.8576340 



7206 3 8576943 
7207*3.8577545 
7208*3.8578148 

7209 8S57875 

7210 38579353 



72113 8579955 
7212,3-8580557 

7213 3.8581156 

7214 3.8581761 

7215 3,8582363 



7^16 3.8582965 
7217i3.8583567: 
7218 3.8584169 
7219,3.8584770 
72203,85&5372 



7221 3.6585973 

7222 3.3586575 

7223 3.8587176 

7224 3.8587777 

7225 3.8588379 



7226,3.6588980 
7227|&8589581 
7228,3 8590181 
722938590782 
7230 3 8591383 



7231.3 8591984 
723238592588 
7233-38593185 
7234 , 3.3593785 
72353 8594385 



7236;3.6a94986i 



7237 
7238 
7239 
7240 



£8595986 
3.8596186 
3.8596786 
3 8597386 



7241 8 8597985 



7242 
7243 
7244 



7196 3.8570912 
7197 18 8571515 
7198;3.8572118 
7199;3.857272 
720013.85733^5 



7245 3.8600384 



7246 
7247 
7248 
724b 
7251 



3.8598585 
3 8599181 
3.8599784 



3.b'C00983 
38601583 
3 8602182 
38602781 
3.8603380 



S96 



LOGARITHMICK 



No. ' Loq:. j)No.| Logr. 



7251 3 

7252-3. 

72533. 
7254:3 
7255*3 

7250 a 

7257|3. 
7258,3- 

7259 3. 

7260 3 



860J9/y;|7301 3 8633823 
8604578| 7302 3.8634418 
.8605177173033.8635013 
8605776 7304*3.8635608 
8606374 1 7305 3.8636202 



.8606973 17306 3.863*6/97 
,8607571; 7807 3.8637391 
•86081701,73083.8637985 
.8608768! 7309 3.8638580 
.860936617310 3.8639174 



726l[3.860y9o4 
72623.8610562! 
7263 3.8611160; 
72643.8611758: 
7265 3 861235 6J 

7206,3.8612954! 
7267;8.8613552j 
7268 3.8614149 
7269,3.8614747 
7270 3.8615344! 



7311 3.8639768 
73123.8640363 
73133.8640956 

7314 3.8641550 

7315 3.8642143 



73163.8642737 
73173.8643331 

7318 3.8643924 

7319 3.8644517 

7320 3.8645111 



727113.8615941 
7272 , 3.8616539 
7273 3.8617136 
( 7274;3.8617733 
7275,3.8618330 



7276 3.8618927 

7277 3.8619524 



7321 3.8645704 
73223.8646297 
7323 3.8646890 



7324 
7325 



7278 
7279 
7280 



3.8620121 
3.8620717i 
3.8621314 



7281 
7282 



7286 
7287 
7283 
7289 
7290 



3.8621910 

3.8622507; 

72833.8623103 

•7284 3.8623699 

72853.8624296 



3.8624892 
3 8665488 
3.8626084 
3.8626680 
3 8627275 



7291 
7292 
7293 
7294 
7295 



7341 



a8627871 
38628467 
3.8629063 
3.8629658 7344 
3.8630253 7345 



3.864748: 
3.8648076 



7326 
7327 
7328 
7329 
7330 



3.8648669 
3.8649262 
3.8649855 
3.8650447 
3.8651040 



7331 

7332 

7333 

73343, 

7335 



3.8651632 
3 8652225 
3.8652817 
1.8653409 
?.8654001 



7337 
7338 
7339 
7340 



7343 



7296,3.8630848 
7297 3.8631443 
729813.8632039 
7299J3.8632634 



7346 
7347 
7348 
7349 



^00-3.8633229 7350 



No. 



7351 

7352 

7353 

73543. 

7355 



Lo*. 



3.8663464 
38664055 
3.8664646 
8665236 
3.8665827 



7356i3 866*417 
73573.8667008 
73583.8667598 
7359J3.8668188 
736013.8668778 



74063, 

7407 

17408 

,7409 

,7410 



7361J3.8669368 
73623.8669958 
7363J3.8670548 
73643.8671138 
7365;3.8671723 



7366 3.8672317 
73673.8672907 
38673496 
3.8674086 
73703.8674675 



7368 
7369 



7371(3.8675264 
73721 3.8675853 
73733.8676442 
73743.8677031 
7375 3.8677620 



73763.8678209 
7377j3.8678798 
7378 ! 3.8679387 



7379 
7380 



3.8679975 
3.8680564 



7381 
7382 
7383 3. 
7384 
7385 



73363.8654593 
3.8655185 
3.8655777 
3.8656369 
38656961 



3.8657552 
7342 3 8658144 
3.8658735 
3.8659327 
3 8659948 



3.8660509 
3.8661100 
38661691 
38662283 
8662873 



7386 
7387 
7388 
7389 
7390 



3.8081152 
3.8681740 
•8682329 
3.8682917 
3.8683505 



3.8684093 
3.8684681 
3.8685269 
3.8685857 
3-8686444 



7391 
7392 
7393 



3.8687032 
3.8687620 
3.8688207 



No. 



7401 
7402 
74033. 
7404 
7405 3, 



7411 
7412 
7413 
7414 
7415 



3.8698768 
3.8699354 
3.8699940 
3.8700526 
3 8701112 



7416 
7417 
7418 
7419 

7420 



7421 
7422 
7423 
7424 

7425 



7426 
7427 
7428 
7429 
7430 



7394;3 8688794 



[395 
7396 
7397 
;7398 
17399 
*7400 



3.8689382 



7431 
7432 
7433 
7434 
7435 



Log. 



.8692904 
.8693491 
.8694077 
.8694664 
.8695251 



,8695837 
18696423 
.8697010 
.8697596 
.8698182 



3.8701697 
3.8702283 
3.8702868 
3.8703454 
3.8704039 



38704624 
3.8705210 
3.8705794 
3.8706380 
3.8706965 



a8707549 
3.8708134 
3.8708719 
3.8709304 
3.8709888 



3.8710423 
3.8711057 
3.8711621 
3.8712226 
3.8712810 



7436 
7437 



3.871S394 

3.8713978 

7438 3.8714562 

,8715146 

3.8715729 



r 439 3. 



7440 



7491 
7492 



7494 
7495 



3.86B9969 
3.8690556 
3.8691143, 
3.8691730J 
38692717 



496 
7497 
7498 
7499 
7500 



3.8745398 
3.8745978 
74933.8746557 
3.8747137 
3.8747716 



No. 



7501 

7502 
7503 
7504 
7505 



3.8751192| 
3.87517711 
3.8752349^ 
3,8752928 
3.8753507 



7506 
7507 
7508 
7509 
7510 



3.8754086 
3.8754664 
3.8755243 
3.8755821 
3.8756399 



3.8756978 
3.8757556 
7513 3.8758134 
75143.8758712 



7511 
7512 



7515 



7516 
7517 



3.8759868 
38760445 
7518(3.8761023 
3.8761601 
38762178 



7519 
7520 



7521 
7522 



7523 3.8763911 



7524 3, 
7525 



7526 3.8765642 



7527 
7528 
7529 
7530 



3.8748296 
3.8748875' 
3.8749454 
3.8750034 
3.8750613 



7531 
7532 3. 
7533 
7534 
7535 3. 



7537 
7538 
7539 



3.8759290 



3.8762656 
38763333 



.8764488 
3.8765065 



3.8766219 

8766796 

(.8767373 

3.8767950 



3.8768526 

i.8769103 

3.8769680 

3.8770256 

,8770833 



75362.8771409 



3.8771985 
3.8772561 
3.8773137 



7540 3.8773713 



7541 

7542 

7543 

75443 

7545 



7546 
7547 
7548 
7549 



3.8774289 
3.8774865 
3.8775441 
8776077 
3.8776592 



3.8777168 
3.8777743 
3 8778319 
3 8778894 
75503 8779470 



ARITHMETICS. 



237 



fe'*' ,- i^g. J >Jo.| Log. IjNo.y Log. 



7651 
7652 
7653 
'7654 
7655 



o.bbo/182 



7551 3.87bU045ft7601 3.8808707 
75523.8780620! 760213.8809279 

7553 3.8781195 760313.8809850 

7554 3 878l770l!7604;3.8810421 

7556 3.8782345 ) 7605*3. .881 0992 
75563\8782919! 7(3063.8811563 .# u56> oa*uul^ 

7557 3.8783494! 7607 

7558 3.8784069:7608 
75595.878464-317609 



3.8837750 
3.8838317 
3.8839885 
3,8839452 



Nx>. 



L<<g. ■ 

7701 3.'8S65471 
770213.8866035 
7703 3.8866599 
7704'3.8867163 
77053 8867V 26 



//U0 •o.otfot^i'G 
7707)3.886885* 
770$!3.88Gl , 417 
7709'3.8869980 



3.8312134!'765?(3 8840580 

3.8812705| ! 7658l3.S84ll54 
/ooy o.c/ e*04-;>j i wy 3.8813276' 7659j-.i>84ir21 7709 3.8869980 
7560 3^8785218! 7^[3^l3847ii76eo!38842£afi 7710 3JJ870544 

7561 3.8785792. 761 l|3.88l4ii 7 766L .j^bS-Ji 77li!5.t87i"f07 

7562 3 8786367. 7612'3.8814988 7662 3.8845421! 7712J3.8S71670 
7563.3 8786941; 7613 3.8815558 17663 3.8843S88 1 771313.8872233 

7564 3.S787515 7614 3.8816128 <7664j3 8844555 7714|3.8872796 J 

7565 3 8788089 :7615 3 8816699 : j 7665 13 88 45122 7715J 3.8S73359 
75oti 3.3788663 76l6 3.8817269 '7666 37884.5688l77i6jj.88/3yy2 



3.8846255 7717 3.8874485 
3 8846821' 771 &3.8875048 
3 8847387! 7719 3.8875610 



75673.8788911' 7617 3.8817840 7667 
7563.3.87892371 76183.8818410 7668 

7569 3.8790385! 7619 3.8818980 17669 o ooi-, o0 / > ( t iw-oot J01U 

7570 3 87.90959 ! 7620 3.88195 50 ; |767() £8847954 7720 3.8876173 

7571 3.87*ji r%:< ■•;! 7621K ««oniorr»/ <•,, i \-i ««TfSto7ilL;cii 3B876736 

3 8877298 

,8877860 
3.8878423 



7571 3.8791532 7621 3.8820120]/ o, 1 13.8848520 \fm 

7572 3.8792106 .7622 3.8820639' 7672 3.8849086' 77*2 

7573 3.8792680J7623 3 # 8821559!;7673 3.8849652i 772 
7574.3.87932531 7624 3.882 1829 7674 3.885021 8 77^ 
7575 3.87938261.7625 ■U822398;l7r?7d .S.8850784 ' 772 

,76*o 3.8822968!'/ o/ o,c oooi350 iTTSj 



75763 8794400 

7577 3.8794973 

7578 3 8795546 

7579 3.8796119 

7580 3.8796692 
?581 3.879/265 
7532 3.8797838 
7583 3.8798411 
75843.8798983 

7585 3. 8799556 

7586 3.8800128 

7587 3.8800701 



?598 3 88Q699„ 
7599 3 8807564 
»7600 3 8808136 



3.8878985 



7727 
7728 
7729 
7730 



7627 3.8823537ii7677 3.8851915 

17628 3.8824107]i7678 3.6852481 

17629 3.3824676!i7679|3.8853047 

7630 3 8825245 ,7680(3,8853612 

76*31 3.8825815l|768i 3.8854178 rTfJi 

7632 3.8826384; i 76823.S854743 :7732'3.8882918 

7633 3 8826953 7683'3.8855308 7733!3.8883420 

•7634 3"' "" ' 

7635 

76*36 3.8828659; 



3 8879541 
3.88G01Q9 
3.8880671 
8881233 
38881795 



3.8882357 



3 8826953 7683 3.8855308 17733 3.8883420 
3.8827522.7684,3.8855874^773^3 8884042 
3 : 8828090i^^85 3.8856439 773 ?3 888^03 



. oat, 



3.8857004,/. o67..hco.uo5 



JNo.l l.o?\ 



775l'3.bbi/oo?7 
7752 , 3.88Q4138 
77533.8894698 
7754;3 8895258 
775513.8895813 



7756 i 5.889£37$ 
7757 3.8896938 
7758-3.889749-8 
7759;3 8898058 
7760 3.8898617 



7761,53839177 
776213.8899736 
7763!3.3900296 
775413.8900855 
776513.8901415 



7700)38901974 , 
7767^3.8902533 
776813.8903092 = 
7769i3.8903651 
7770 3 8904210 k 



. . _ . 7637 3.8829228i!768/ 3.8857569 7737 
75£8 3.8801273 7638 3.8829797. {7688 3.88581347738 
7589 3.8801846 7639 3.8830365] 76$) 3.8858699 7759 
7590 3.880241 8 7640 3.8830934 : 7690)3 8859263 7:40 
' 7591 3.8to2990|7641 5.8831502 I ^0^1 
7592 3.8803562 7642 3.883207017692 
7593 3.8804134! 7643 38832639 ;7693 
75943.8804706 7644 3.88S3207'7694 
7595 3.8805278J764.S 3.88^3:75 '7695 
7^^63.88058501 7o46 3.883-KvO j76{:6 
7597 3 8806421J7647 3.883491! |7697 



3.8859^28 77-ii , 

•8860393 17742 3.8888532: 
3.8860957 7743 
5.8861522 7744 



3.8885726! 
8886287J 
3 8886848 
3.8887410! 
3.8887971 



777l!3.b904769 f 

777213 8905328 

777313.8905887 

7774 ! 3.8906443 

7775 ! 3.890700 4 

7776^.^V?56S 

7777^3 S908121 

7778J3.8908679 

7779;3.8909238 

7780^^22212? 
77Hi ^.o»iu J54 
7782.3 8910912 
7783|3.3911470 
7784:3 8S 12028 
7785' ? 89125 66 
//S6«> 8^13144 
7787!3.8913702 
7788:3.8914259 
7789|3.8914ei7 
7790 3.8915375 



7791i^.^lo932 
7792'3.8916489 
7793|3.8917C47 

- - , 7794:3.8917^04 

3.8862086 ;7745H 8890214' 7795 3 - C U»61 



3.8889093 ' 



3.8889653 



3.8d6205l 774G|j.o6jU/7^ 779o-5.:» • i=/li> 
. . I..S863215 i7747i3.88 ( J1336 : 779713 8<)J:«2?5 
7646 3.8835479 7698 3.8863779 1774! 13.8891896^ 7798 
764S3.8836047,f7699S.3864S43 1774V 3.8892457i77i>i> 3.8920389 
650 J.8836614. j 7700[3.88649O7 17750 '3.88930171 7«00 3.8920S46 



SI 



•2SS 



LOGARITHMICK 



No I L^fr. 



7fi0li3.8929503j 
780213.8922059 
7803;'3.8926616j 
r80*!3.893317i' 
7805J3.S923729 

■80S'3.8924285' 
7807|3.8924842 
7808^.8925398 
7809|3.8925954 

7810 38926510; 

7811 3.892706^ 
8123.8927622 
813 3.8928178 

7814 3 8928734 
7815,3 8929290 
78163.8929846 
7S17I3 8930401 
8133.8930957 
7819 3.8931512 
78203.8932068 



7821 3 8932623 
7822.3 8933178 
78233.8933733 
7824^3.8934288 
7825 3 8934843 

7826 ! 3.8935398 
7827,3.8935953 
78283.8936508 
7829J3 8937063 
7830,3.8937518 
7*31 3 8938172 
8323.8938727 
78333.8939281 

7834 3.8939836 

7835 3.8940392! 



7901)3 
790213 
7903'3. 
7904'3. 
7905J3 
790&3. 
790713 
79083 
79093 

,79103 

/&613 8954778 ! '7911J3. 

7862 3 8955330! 7912J3. 

7863 3.8i>55883! 7913i3. 
78643.8956435 7914fc? 



7865 3 89569 87 
7666 3.8957539 

7867 3.8958092 

7868 3.8958644 
78693.8959195 
7870 3 8959788 



'&>o3.8y40944! 
7837 3 8941498! 
78383.8942053 
7839 3.8942607| 
"840 3.89431611 
7 841 3.8943715 

7842 3.8944268 

7843 3.8944822 
78443,8945376 
7 8453.8i : 45S2P 
7 r o46'3T89i6583 
7PA7 18947037 
T 8-:G3 894759C 
T 84^3 8948.14,: 
7850 3 8944597 



No. | Lour. 



No. i Lot*. 



78J 13.8949250 
78523.8949803 

7853 3.8950356! 

7854 3.8950909! 

7855 3 8951462 ' 
/bob 3.89520 13! 
7857 3.8952568 1 
78583 8953120, 

7859 3.8953673; 

7860 3 8954225 



8976821 ; 
8977370 
8977920 
8978469 
8979019 
8979578 
.8980117 
.8980667 
8981216 
8981765 



78713.8960299 
7872 3 8960851 
7873.3 8961403 
7874 3.8961954 
78753 8962506 
? tt/ 6 3.8963057 
78773.8963608 
78783.8964160 
7879 3.8964711 
788038965262 
/881 3.8965813 
7882 3.8966364 
78833.8966915 
78843.8967466 
7885 3.8968017 
7bo6 3 896856*5 
78873 8969118 
'8883.8969669 
78893 8970220 
7390 3.8970770 



7915 3 
79163. 
7917J3, 
7918 ! 3. 
79193. 
7920*3 



.8982314 
.8982363 
.8983412 
,8983960 
8984509 
.89850~58 
.8985606 
,8986155 
.8986703 
8987252 



7922 



7921jo.8yb7800 
.8988348 
.8988897, 
.8989445, 
.8989993! 



7891 3.8971320 
78923.8971871 
7893'3 8972421 
8943 8972971 
(895 3j&7352\\ty 
rs?6 3.8974071 
78973.8974622 

7898 3;8975171 

7899 3.8975722 
79003.8976271 



7923 

,'7924 

7925 

7926 

7927 

7928 

7929 

7930 

,7931 

'7932 

J7933 

7934J3. 

79353, 

79363 

79373 

179383 

79393 

79403 



8990541 
8991089; 
.8991636, 
8992184J 
8992732! 



.8993279 
.8993827 
.8994375 
.8994922 
.8995469 
.8996017 
89965641 
8997111 
8997658 
8998205 



i794l;3. 
7942 ! 3, 
79433 
7944 3 
945 3. 
79465" 
79473. 

7948 3 

7949 3. 

7950 3. 



8998752 
8999299 
8999846 
9000392 
^000939 
9061486 
9002032 
9002579 
.9003125 
, f J003671 



> u.l Lot*. 1 
"951 '3300421-8 
79523.9004764 

7953 3 9005310 

7954 3.9005856 
7955J3 9006402 
/y5"6 ! 3.9006948 
7957J3.90074P4 

7958 3.9008039 

7959 3.9008585 
7960*39009131 
7961 ! 3 9009676 
79623.9010222 
7963 3 9010767 



7964 
7965 
7966 
7967 
7968 
7969 
7970 



7971 
7972 
7973 



3.9011313 
^9011858 
3.9012403 
39012948 
3.9013493 
3.9014038 
3.9014583 



8U01-J 
80023 
80033 
8004'3. 
8005'3. 



9031443 
9031985 
9032528 
.9033071 
9033613 



8006 o. 
80073 
8008 3, 
m09'3. 
8010 ! 3, 



9015128 
3.9015673 
3.9016218 
7974J3.9016762 
797513^016307 
79763.9017851 

7977 3.9018396 

7978 3.9018940 

7979 3.9019485 
7980*3 9020029 
7y«l 3.9020573 
7982 ; 3 9021117 
798313.9021661 
7984o.9022205 
7985 3 902274 9 
7986*3.902329; 
79873.90238371 

7988 39024381; 

7989 3.9024924' 

7990 3 9025468 
799i 1 3.90260li l 
7992^3-9026555 
7993 3.9027098 
79943.9027641 
7995f3.9023185 



801 H3, 

,80123 

80133. 

8014';*. 

801513 

8016*3 

80173 

8018 

801913 

8020 ! 3. 



i>Oo4l56 

9034698 

9035241 

9035783 

9036325 

.9036867 

9037409. 

9037951 

.903849 

9039035 



.9039577 
.9040119 
9040661 
9041202 
.9041744 



8021.o.yu42285 
802213.9042827 
8023(3.9043368 

8024 3 9043909 

8025 3 904445 

8026 3.9044992 
8027J3.9045533 
80283.9046074 
8029!3.9046615 
8030 ! S 9047155 



40313.9047696 
603213.9048237 
80333.9048778 
8034 l 3.9049318 
8035 ! 3.9049859 



7996!3.yUXi8728; 
79973.9029271' 
7998^.9029814 
7999 3.9030357 
7000 ! 3.9030900 



«036 3.9050393 
8037)3.9050940 

8038 3.9051480 

8039 3.9052020 

8040 3.9052560 



8O41'3.y053101 

8042 3.9053641 ' 

8043 3.9054181 
80443.9054721 
8045 3.9055260 



80463.905^800 

8047 3.9056340 

8048 3.9056880 

8049 3.V057419 
80503.9057959' 



ARITIIMETICK 



• 339 



No. Loir. 



8U51 
8052 
8053 1 
8054 
8055 



3.9056496 
3.9059038 
3.9059577 
3.9060116 
39060655 



8U5o 

805! 

8058 

8059 

8060 



3.9061 la j 
3.9061734 
19062273 
3.9062812 
9063350 



8061 
8062 
8063 
8064 
8065 



80o6 
8067 
8068 
8059 
8070 



8071 
8072 

807; 
8074 
8075 



8101 3.9085*86 
81023.9035922 
8103 3.9086458 



8104 
3105 



81uo 3 9088066' 8156 3 9114772 
8107 39088602 1 8157J3 9115305 
8103 3.90891371 81583.9115837 

8109 3 9089673: 8159 ! 3.91 16369 

8110 3.9090209J 8160:3.9116902 



3:9063689. 
39064428 
9064967i 
3.9065505 
3 9066044' 



8111 
8112 
8113 
8114 
8115 



3.9U66582 
3 9067121 
3.9067659 
3.9068197 
3.9068735 



8116 
8117 
8118 
8119 



9120094 
9120626 
9121157 

,9121689 

8120|3 90955601 8170'3 9122221 



9069273 
3.9069812 
3 9070350 
3.9070887 
3.9071425 



8121 
8122 
8123 



8076 
S077 
8078 
8079 
8080 

8081 
8082 
8033 
8084, 
8085 



3.9071963 
3.9072501 
3 9073038 
3.9073576 
3.9074114 



8086 
8037 
808S 
8089 
' 8090 

8091 
8092 
8093 
8094 
8095 



8096 
8037 
8098 
8099 
810C 



3.9074651 
3.9075188 
3.9075726 
3.9076263 
3.9076801; 



39077337. 
3.90/7874 
3.9078411 
3.9078948 
3.9079485 



3.9080022 
3 9080559 
3.9081095 
3.9081632 
.9082169 



3.90827U5 
3.9083241 
3.9033778 
3 9084314 
3 9084850 



No.| L<ȣ. IN'). 



3.9086994 
3.9087530 



Lot;. | No.j Lov^. 



8151 

8152 

8153! 

81543 

81553 



.911210* 
9112642 
,9113174 
.9113707 
9114240 



3 9uyo/44;8i6i:3. 

3.9091279j 8162i3. 
3 9091815J 8163 3, 
3.9092350; 8164*3 
3 9092885 8165'3 



,9117434 
,9117966 
,911849b 
9119030) 
.9119562 



3 9093420 8166 3.5 
3.9093955' 8167J3.1 
_ 9094490';8168 3.1 
3,9095025,!8169 3! 



3.90960y5--8l7l3 
3.9096630."8172 3 
3.9097165 8173)3. 
8124(3.9097599 i$l74 t 3 < 
8125J 3.90.98234 8175J3. 
8126&.9098768 8176i3 
8127|3.9099303 .8177)3 



9122752 
9123284 
,9123815 
.9124346 
9124878 



9125409(82263. 

.9125940 ;8227 3 
8128|3.9099837 '81783.9126471 18228 
8129 3.9100371 8179 3.0127002 j822<) 3 
8130'3,9100905 8180 ! 3 9127533 [3230 3 



8201 3.9138668 
82023 9139198 
82033.9139727 
32043.9140257 
8205 3 9140786 



8206 3 9141315 
82073 9141844 
8208S.9 142373 
82093.9142903 
8210 3 9143432 

621 1 o. 

8212 3 

8213 3. 
82143. 
8215 3 



8210 3 
82173 

8218 3 

8219 3 
82203 



9143^01 
9144489 
.9145018 
9145547 
,9146076 

91406U4 
9147134 
.9147661 
.9148190 
.9148718 



82213 
82223, 
8223 3 
82243 
8225 3 



,9149246 
9149775 
9150303 
9150831 
915135V. 



8131:3.9101440 j8181'3.9128064j 
3132 3.9101974 8182'3.9128595j 
813SJ3.9102508 >8m 3.9129126: 
8134 ! 3.9103042 '81843 9129656 
81353.9103576 J8185 3 9130187 



9151887 
9152415 
.9152943 
9153471 
9153998 



81363.9104109 
31373.9104643 
81333.9105177 
813939105710 
81403.9106244 



8232 
8233 
8234 
8235 



82371.1 



8186 3.9130717j!8236 

18187 3 

|8188 

J81893 

8190 3 



:.0131248i 
5.91317781 



8238;3 
9132309^8239^ 
9132839: 



82313915452fc 
9155054 
.9155581 
.9156109 
■9156636 

.9157163 
9157691 
9158218 
.9158745 
915927:2 



8240 3 



8141 3.91U6778 

8142 3.9107311 

8143 3.9107864 
81443.9108378 
8145 S 9108911 



'8191 3 
|81923 
18193 3, 
[81943 
J8195 3 



,91o33b9j 
9.338991 
,9134430! 
9134960i 
9135490 



1463.9109444 81963.9136019 
.9136549 



S147 ; 3.9109977 
81483 9110510 
8149J3.91 11043 
8150 ! 3.9111576 



819713. 

81983 
8199J3. 



H24i 3. 

8242 3. 

8243 3. 

8244 3. 

8245 3, 



.9ioy/V_ 
9:60326 
,9160853 
.9161380 

.9161907 



8246 3, 
18247 "3 



8200' 



913707^,8248 
.9 137601- 113249 3. 
913S139 I! 3250:> 



9id243o 
91629 (Vj 
'.'16343; 
916401:; 

916453'- 



N< 



L«ir. 



8251 3.9165066 
82523-.9165592 
8253j3.9166118 
82543.9166645 
8255 3 9167171 



82563.910? 697 
8257-3.9168223 
82583.9168749 
8259 3.9169275 
8260'3.9169800 



6261 o yi/UJ26 
8262'3.9170852 
8263 , S.9171S78 
8264 ! 3.9171903 
'8265 ! 3 9172429 



<«26t> 3.91/ 2954 
I8267J3 9173479 
,8268 3.9174005 
S8269 3.9174530 
'8270 3 9175055 



i8271 J.9175. 10 
8272 3.9176105 
82733.9176630 
82743.9177155 
8275 3 9177680 



82763. 
;82773 
J82783. 
•8279 3. 
S280 3 



>J178205 
,9178730 
,9179254 
9179779 
936030. 




^160*628 
9181352 
9181877 
9182401 
9182925 

9 i 634*19 
913S973 
918449? 
9185021 
,9185545 



'biib'l •> 
182^2 a 
8<?93 s 
'S294? 

'32C5 3 

S296o 

k^973. 
: J 29V 3. 
! ->299 3 

133002, 



b-j 6(5069 
9186593 
,9187117 
9187640 
9188164 



vl «868; 
.9189211 
91 897^4 
9190258 
,9190781 



240 



LOGARITHMICK 



No.i Lop- If No. L<>£. |M ,J 'I L'»*r. IjNoJ Lo*. if No 

..^. ".„ 7TTTT ■•» .-..- '-*. ~ . . . * .. . I [T. , . .~ — ""... t — - 



83Ui<3 9191;>U*:83513 
8302|> ^19l8-i/( 8352 3 
8303|3 91J235018353 3 
83043 9192873 835413 
8305 3 9193396,8355 



83 jo 3 919391* 'ttj do 3 yjf99b-±: 
830713 <U9441*i -.8357 3 9220J04; 
83083 919496o 3358 3 9221024! 
8309(3 9195488 J83.J9 3 9221543! 
831T3 9196010!J8360!3 9222063' 



83ib;3 
8317 ! 3 
83183 
8319 3 

ill:" 

8v^; r 

83223 
8323 3 
8314 3 

83?53 



92l7o85*b4Ui 3 9^433 lu!!84ol!3 
9217905 8402 3 9243827||8453i3 
9218425 8403 3 9244344! 8453 3 
9218J45J8404 3 9244860*8454 3 
9219465! 8405 3 92453771 8455 3 



Lo^-. 



9269U8i 8501 
9269595^8502 
9270109 (8503 
9270622;i8504|3 
927113fr;8505i3 



9294700 
9295211 

9295722 
9296233 ^ 
9296743 



8406*3 
840713 
8408*3 
34093 
84103 



i>2*o8?4 '8456 3 
92*6410;!8457<3 
92469271184583 
<X247444 l -8459 3 
9247960' 8460 3 



927lao0!;8506jJ 
9272163 8507 3 
9272677i!85083 
9273190;8509;3 
9T73704 ! :S510!3 



9^97254 
9297764 
9298275 
9298785 
9299296 



8jli;3 yrJ65Jo| 8361 3 

8312;3 9157055 !8362'3 . .._._.„ 

8315;3 9197578|!83o3!3 922362l||8413'3 

4314*3 9198100l!8364'3 

8J15;,5 9198633JJ8365S3 



^2^2562.:8411!3 
9223102 ! i8412|3 

I* 

9234140 [8414 , 3 
9224659 ;8415 : 3 



*J24cJ4/6:«4ol3 

9248993;'84623 

6249506^8463 3 

9250025 8464-3 9275757JJ85143 

9250541- ,8465 3 92762701 |35I5|3 



J742l/j,8511j3 
9274/30' 851213 
9275343! '8513:3 



i>299806 
9300316 
9300826 
9301336 
9301847 



9199145 ;83oof3 
9199667,183673 



92001891 
92.10711 
9'-201?3.1 

ylsOlf'aJ 

9207177 
9202799 
9203321 

92 J3342 



J83683 
*836^j3 
1837013 



9^251/9*18416^ 
922569£8417i3 
9226217! 8418 ! 3 



9226736 



y251057<:3±663 
9252572, |8467'3 
9252089i l 84683 
9252605 |84693 
92S7255 ,! 8420'3 f253121 ! |8470'3 



8419!3 



9276/83;!8516 3 
9277296}I85173 
9277808! I8518I3 
9278321j ! 8519!3 
9278834! !8520 3 



«?302357 
9302866 
9303376 
9303886 
9304396 



|8371J3 0^27773*8421.3 



; 8372|3 
! 8373 : 3 
'8374 ! 3 
.83753 



9228292.84223 
922881li!8423 3 
9229330 , ;8424 , 3 
9229848' !8425 i 3 



92536371-8471 3 
9254l52l ! 8472 ! 3 
9254668'8473;3 
9255184;.8474 ; 3 
9255699'i8475i3 



8826 > 92043o4 i83. 5 
8327,3 9204336 m77\3 
8328 3 9205407 '83783 
8329;3 9205979 '83793 
83303 920645 •;) 8380!: 



83.1 ;3 92065V 1 
83323 620:493 
8333:3 9203014 
8334,3 9208535 
833513 0209056 



8336 3 
833713 
8338 3 
83393 
8340 3 
8341:3 
83453 
4343|3 
83443 
834513 



♦.-2095 

9710008 

9210619 

9211140 

9211661 

1512181 
9212702 
9213229 
921.374.' 
921456' 



8346|3 
834713 
834813 
3349fs 
83503 



92147-54 

921530^ 

921582 

921634.' 

9216865 



9.^30oo/i8426;3 
7230885 ! |8427j3 
9231404' 84283" 
923192SJi8429'3 
9232440 18430 ! 3 



92/93471I8521J3 
9279859!85223 
9280372-8523 
928088S|l8524 
9281397 ',8525 3 



9256215 8476;3 

9256730^8477(3 
9257245:184783 
9257761 ![8479!3 
9258276 !8480|3 



18381,3 
8382!3 
133833 
8384;3 
838513 

,83St>3 

8387 3 

8388 3 

8389 3 

8390 3 



9232958 8431 3 
9233477'i8432.3 
923.3995 '8433 3 
9234513 84343 
92 ^5031 8435 3 
923 354^ 8436 3 
9236066 84373 
9236584 8438 3 
9237102 8439 3 
9237620 8440 3 



9258791|i8481.3 

9259306 J8482I3 

925982l'84833 

9260336i;8484l3 

9260851 118485 3 

926l366:'8486 

9261880; ; 8487 

9262395!!8488 

9262910!8189 

9263424] 8490 3 



9281909-jj8526 
9282422:8527 
9282934II8528 
9283446J8529 
9283959! i 8530 



i83$i 3 

'3392, 
8393 
18394 
'8395 



i8390 
13397 
I8398J3 

18399 
; 3400 



923813/84*13 
9238655 84423 
9239172:8443 3 
92396^0'84443 
9240'07'84453 



9263939 .8491 
9264443 8492 
92*4958 849 
9265482 8494 

9265997 8495 

"9240724'84463 9266511 ;84% 
9241 242; 8447 3 9267025:8497 
9241759;.84483 9267539 i- 1 849 8 
9242276' 84493 926805318499 
9242792*18450 3 9268567' =8500' 



9284472! 
9284983J 
9285495 
92860071 
92 86518 1 

9287030! 
92S7542 ! 
9288054 
9288565 
928907! 



8531 
,8532 
i8533 
8534 
8535 



9289588 
9290100 
9290611 
9291123 
9291634 



9*92145 

929265C 

3 9293167 

3 9293678 

.9294189 



8536 
8537 
8538 
! 8539 
8540 



8541 
8542 
854 
8544 
8545 



8546 
8547 
3548 
8549 
8550 



9304906 
9305415 
9305925 
9306434 
9306944 



9307453 
9307963 
9308472 
9308981 
9309490 



9309*99 
9310508 
9311017 
9311526 
9312035 



9312544 
9313053 
9313562 
9314070 
9314579 



9315087 

9315595 

9816104 

9316612 

9317121 

3 9317629 

3 9318137 

3 9318645 

3 9319153 

3 9319661 



ARITHMETICS 



**l 



8551 

8552 
8553 



No 



8556 

855/ 

855& 

855'. 

8560 



L:>%. | No. Log | 



393A>169i 
£9320677] 
3.9321185) 

8554 3.9321692! 

8555 3.9322200! 



3.934546^' 

3.9345994 

3.9346499 

.9347004 

8605J3.934750'' 1 



8601 
8602 
8603 
8604 



3.9322708! 

.9323215J 



86J6!3.95480i;< 



3.932373& 

'•1932423a ! 



9,3. 



8607 
8608 
8609 



393247381 



86103 9350032 



8oDli3.'J3i5,243! 
896213.93257521 

8563|3.9326259| 
8564 ! 3.9326767! 
8565 ft /?327274 ! 

85oO;3.9527/bi: 
8567J3.9328288' 

8568 3.9328795! 

8569 3.93293011 
85703.9329808' 



8611 
8612 
861, 
8614 



8616 
8617 
8618 
8619 
8520 



8571J39330315 
8572(3.9330822 
8573j39331328 

8574 3.9331835 

8575 3.9332341 



8621 



85763 9332&43 
85773.9333354 
3 9333860 
3 9334367 
3.9334873 



8578 
8579 
8580 



8581 
8582 
8583 
8584 
8585 



.9336391 
3.9336894: 
3.9337403" 



8586-3.9:i3/9U9, 

8587 39338415 

8588 3.9338920' 

8589 3.9339426 1 
8590 ! 3.9339932' 



8591 3.93*0* j/; 

8592 3 9340943: 
85933.934 1448; 
85943.93419531 
859.5,39342456 



85yo; 3 9342^64! 
!8 197:3 9343469 
!35:>8j3.9343974| 
859!) 3 93444791 
!8590'3 93449851 



boo! 3,93 /0o63 
86523.9371165 
8653 39371667 
86543.9372169 
8655 3.9372671 



3.9348518 

."1.9349023 

.9349527 



86ib'3.93731/2 
8657S.D373674 
3558 f 3.93/4176 
8659-3.9374677 
3660 3 9375179. 



3.9350050 
3.935 1040 
3 9351544 
3.9352C4J; 
86153.9352553 



8661 3.9375680 

3662 3.9376182 

3663 3.9376683 
8664c.')377184; 
8665 3.9377686 



3.935^05/ 
3 9353561 

3.9354065 
3.9354569 
3 9355073 



3 9355570 
8622J3.9356080 
3623 >*9356584 

8624 3 9357087: 

8625 3 9357591 



8627 
8628 
8629 
86303. 



3 9358U95 
3.9358498' 
3.9359101 
3.9359605, 
;.93601Q8 ! 



3 93355/9; tfo3ij3.93o061l! 

3.93358851 3632!3.9361114; 
8633l393(51617i 
8634J3-9362120 



3635=3.9362623 



3636 3.9363120 
3637J3 9363629 
36383.9364132 
8639 3.9364635 
86403.9365137 



86a'3.9365640i 
36423.9366143; 
3643J3.9366645! 
3644 ! 3.9367H8 ! 



86463.9368152 
8647(3.9368655 
86483.7369157 
8649J3.9369659 
8650!39370161 



IV -j.\ 



l!i\l 



8701 3»y395o92!'875l 



87023.9396191 

8703 3.9396690 

8704 3-9397189 
8705J3 9397688 



8752 
8753 
8754 



Lop;. 



3.9420577 
39421073 
3.9427569 
3.9422065 



8755;3.9422562 



86663 9378187 
866739378688 
86683 9379189 

8669 3.9379690 

8670 3. 938019 1 

8671 3.9380692 
86723.9381193 
8573 39381693 
86743.9382194 
8675 3.9382695 



3676 3.9383195 
8677 3.9383699 
86783 9384196 
86793 9344697 
86803 9385197 



8081 3 9385698 
3682 3.9386198 
86833.9386698- 
36843.9387198 
8685 39387698 



8686 3 9388798 
36873 9388698 
36883.9389198 
86893.9389698 
86903.9390198 



8691 3.9390697 



8692 
8693 
3694 
8695 



3.9391196 
3.9391697 
3.9392196 
3.9392696 



3696 
697 



3.9393195 
39393695 
S69S;^394194 
869913.9394693 
37003.9395193 



8706o.btt98187 ;b/a6|o MU3\)5& 

8707 3 93G8685 87o7|3 9423553 

8708 3 9399184! Hi 58 3.9424049 

8709 3.9399683 : 8759 3.9424545 
8710 3940018':! 8760 3 9425041 

3.9425537 
3.9426032 
3.9426528 
3 9427024 
39427519 



8/11 3.9*1/0680' j 8/ 61 
87123.94011/9|j8762 
8713!3 9401677|I8763 
8714'3.940217(»jb'764 
8715'3 9402674 < ,87 63 

37163.9403172' 8700 



8717,39403670; 
8718 3.9404169 
87193.9404667! 
8720:3.9405165: 



87213 9405663 
8722 : 3^40616l' 
8723 ! 3.9407659i 

8724 3.9407157! 

8725 3.9408654 



8726 3.9408152 
8727i3.9408650 
872813.! 
8729|3.9409645' 



*!« 



8776|3, 

mm 

•9409147| 8778 3 
8779 
8780 ! 



8730'3.9410142 



8767 
8768 
8769 
8770 



8771 

8772 

877 

8774 

8775 



3 9428015 
39428510 
3.9429005 
3.9429501 
3.9429996 



> 9430491 
5.9430986 
1.9431481 
S.9431976 
) 9432471 



8731 
8732 
873S 
8734 
8735 



,9432966 
9433461 
.9433956 
.9434450 
9434945 



:.9410640;;878i;3.9435440 
:.9411137j|8782,3.9435934 
:.9411635ij8783 39436429 
;.9412132![8784 3.9436923 
! 9412629i 8785 3 9437418 



8736 
8737 

8738; 

87393, 

8740'3 



94131261:8786 3.9437912 
,9413623: : 8787 3 9438406 



9414120 
9414617 
9415114 



8741i3.94l5611 
8742*3.9416108 

8743 3.9416605 

8744 3.9417101 

8745 3.9417598 



874613 9418095 
8747 39418591 
3748 39419038 

8749 3.9419584! 

8750 3.94200811 



8788 39438900 

8789 3.9459394 

8790 3 9439889 



8791 3:9440383 
87923.9440877 
8793 3.9441371 
87943.9441865 
8795 "3 9442358 



8796o.9442852 



8797 
8798 
8799 
8800 1 



3 9443346 
3.9443840 
3.9444333 
3.9444827' 



S46 



LOGARITIIMICK 



No 



8801 

&802 

8803 

8804 

8805 

880o3 

3807 

8808 

8809 

8310 



L*>£. 



,No. 



jOiX 



;.9445J20,885lfc 
.94458i4 ( 8852 3. 

.9446307, 8853'3.^. . ^^ „.^„„^ 

.9446800 8854J3.9471395, 1 8304 3.9495852 
.9447291- |8S55!3.9471886j 8905 3 9496339 



94699231 
9470414 
9470905 



.No., L<K> 



89J1 3.9494388 

8902 3.9494876 

8903 3.9495364 



No. | 

SSI's" 

89523, 
89533, 
8954'3. 
8955 3 



.9447; 37,88^13 9472376.^^06 3.9496827j 
8857 39472866 8907 3.9497315; 



.9448280 
9448773 
.9449266 
9449759 



88583 
8859 ; 3.S 



9473357! 

,9473847; ! 

8860[3.9474337 : 



88113.9450J52 
681? 3.9450745 
8813:9451238 
8814(3 9451730 
881513 9452223 



8861 3.94? .382/. 

8862 3.9475317, 



8310.3.9452/ lo 
88173.9453208 
8818'3.9453701 
8819 3.9454193 
88203.9454686 



^ 



8908 3.9497802 

8909 3.9498290 

8910 3 9498777, 



8863 



3.9475807 



88643.9476297J 



8865 



39476787 



8366 
7867 
8868 
8869 
8870 



3.9477277i 
J9477767; 
39478357; 
3.94787471 
3.9479236 



88213.9455176 
882239455671 
8823 3.9456163 
882439456655 
88253.9457147 



8871 
8872 
8873 
8874 
8875 



88263.9457639 
88273.9458131! 
8828 3.9458623 
8829|3.9459115 
88303.9459607, 



8877 
8878 
8879 
8880 



89563. 
8957t3. 
8958'3. 
8959J3, 
8960 ! 3 



8911 3.9499264 

8912 3.9499752 

8913 3.9500230 

8914 3.9500726 

8915 3.9501213 



8916 3.9501701J 

8917 3.9502188 

8918 3.9502675; 

8919 3.9503162 

8920 3.9503649 



3.9479726 
3.9480215 
3.9480705 
3.9481194 
3.9481684 



89213.9504135 
89223.9504622; 
89233.9505109, 
89243.0505596 
892539506082; 



88763.9482173 
3.9482662 

88783.9483151 

.9483641 

3.9484130 



8831 3.94t>0uyy, 8881 
8832 39460591:8882 
8833.3.9461086; 8883 
8834 ! 3.9461574 ! 8884 
8835 3 946206 618885 
8836'3 9462557||8886 
883713.9463048,8887 
88383 9463540 1 8888 
8839j3.946403l||8889 



3.9484619 
3.9485108 
3.9485597 
9486085 
3.9486574 



3.9487063 
3.9487552 
3.9488040 
3.9488529 
8840'3 9464523| ! 8890 3.9489018 



8'34ll3.9465014 
t8842 ! 3.9465505 
884313.9465996 
8844 ; 3.946f>437 
8845|3.9466673 



(8891 
18892 
3893 
3894 
8895 



8846: 3.9467 40^ 
8347,3 9467960 
884813 9468451 ' 
88493.9468942 
88503.9469433 



3.9489506 
3.9489995 
3.9490483 

3.9490971 j 
3.9491460 



8961(3.9523565: 
8962J3.9524049; 
8963 ! 3 9524534* 
8964 f 3 9525018! 
8965j3. 95255Q3 

8966'3. 
896713. 
896813 
8969*3 



8926 3.9506569 
89273.9507055 
89283 9507542 
8929 3.9508028 
89303.950851 5 

89313.9509001 
89323.9509487 
8933'3.9509973 
89343.9510459 

8935 3.951094 6 

8936 3.9511432 
8937JS-9511918 
39383.9512404 
89393 9512889 
8940'3.9513375 



8941:3 9513861 
8942,3.9514347 
8943-3.9514832 
39443.9515318 1 
8945.3.9515803 1 



8397 
8398 
8899 
390' 



3.9491948J 
3.9492436 
3.9492924 
39493412 
3.94939001 



Lo^. 



9518710 
.9519201 
.9519686 
.9520171 
9520656 



9521141 



No. 



90013 
9002 ! 3. 
k>003'3, 
!9004'3 
J9005 3. 



Log. f 

35429081 
9548390 
,9543873 
9544355 
9544337 



19006 £ 



.9544819 

i , 521626'j9007j3 9545302 

i.952211l!!9008 i 3 9546284 

' 9546766 

9547248 



.9522595, |9009;3.< 
9523080; 9010 3 < 



9011 3.9547730 

9012 3.9548212 

9013 3.9548694 
901413.9549176 

9015 3 9549657 



4970 



8971 
8972 
8973 
8974 
8975 



8976 
8977 
8978 
8979 



8981 
8982 
18983 
8984 
8985 



y525987|j9U16 

9526472 J9017 

952695619018 

9527440' ! 9019 

'9527924 ;: 9020 

.9528409'J9021 

1.9528893 ;9022 

.95293771 902, 

.9529861; 

.9530345 



3.9550139 
|3.9550621 
;3.9551102 
39551585 
39552065 



3.9530828. 
3.9531312 
3.9531796' 
3.9532280! 
898013 9532763 



39533247 
3.9533751 
3:9534214 
3.9534697 
3.9535181 



898639535664 
898713.9536147 
8988'3.9536631 
8989i3.9537114 
8990139537597 



3.9552547 
'39553028 
39553510 
J39553991 
39554472 



9024 
9025 

902613.9554953 
902713:955543 ' 
9028 3.9555916 
9029i3 9556397 
9030 ? 3.95568 78 

9031(3 9557358 
9032 3.9557839 
9033 ! 3.9558320 
90343.9558801 
903539559282 



8991;3'9538080 
'8992J3 9538563 
899313 9539046 
8994!3;9539529 
8995|3 9540012 



8946;3.9516289j 
89473.9516774 

8948 3.9517260 1 

8949 3.9517745] 
8950I3.9518230! 



8996'3 9540494 



190363.9559762 
9037:3.9560243 
£038 39560723 
19039 39561204 
9040 3 9561684 



899; 
8998 
8999 
9000 



3.9540977. 
3-9541460 
3-9541943, 
3.9542425 



19041 3.9562165 
'904213.9562645 
|9043 3 9563125 
t9044!3.9563606 
|9045| 3 9564 086: 

19046 3.9564566 
\9047\3 9565046 
i 9048,3 95655i6\ 
, 9049 3.956600& 
1905013.9566486 



ARITHMETIC*. 



243 



«No. Log. 1 

90513.9566966 

9052 3.9567445 

9053 39567925 
90543-9568405 
9055 39568885 



905£'3.9569364 
9057J3-9569844 
9058*3.9570323 
905939570803 
90603.9571282 



9063 



906439573199 



9065 



9067 
9068 
9069 
9070 



906113.9571761 
9062 3.9572241 



3 9572720 



3.9573678 



9066 3.9574157 



.9574636 
3.9575115 
3.9575594 
3 9576075 



9071 
9072 
9073 
9074 
9075 



39576552 
3.9577030 
3.9577509 
3.9577988 
3.9578466 



9076 
9077 
9078 
9079 
9080 



3.9578945 
3.9579423 
3.9579902 
3.9580380' 
3.9580858 



908l|3 9581337j 
9082'3.9581815! 
9083|39582293; 
9084'3958277l! 
9085J3.6583249' 



908613.9583727, 
9087 3 9584205 
90883.9584683 
9089;39585161 
90903 9585639 



No., 

or 3 

.02*3, 
03 3. 
04'3, 
053 
063 
07]3, 
083, 
09 ! 3 
103, 



No., Log. | No. | Log. j 



909i;3.^8oll6 
90923.9586524 
9093'3.958r072 ., 
909439587549 \g 
9095.3 9588027; 



90963.9588505 
9097(3.9588982 9 
90983 9589459, ]9 
909913 95S9957II9 
9100<3 9590414 ,| 9 



9590891 
.9591368 
9591845 
.9592322 
9592800 



11|3. 
123. 
13|3. 
143. 

153. 

163^ 
17|3. 
183, 
193. 
20 3, 



.9593276 
.9593753 
.9594230 
.9594707: 
.9595184 

.9595660 
.9596137 
9596614 
,959) 090 
,959756/ 



91513.9614680 
9152'3 9615160 
91533 9615635 
91543 9616109 
91553.961658 3 
91563.^617058 
9157J3.9617532 
9158'3.9618006 
915913.9618481 
9160L3 9618955 



.9598043 
,9598520 
,9598996 
,9599472 
.9599948 



9106 
9167 
•168 
6169 
9170 



2i;3. 
223, 
23 3. 
243. 
25 S. 



9600425 
9600901 
9601377 
,9601853 
,9602329 



26|3. 

ar% 

283. 

29 3, 
30'3. 



.9602805 
#603281 
9603756 
.9604232 
,9604708 



313 

32i3. 
33J3. 
34 j 3. 
353 
36|a 
373. 
383. 
393, 
40 3. 
3~ 
3 



9605183 
9605659 
,9606135 
,9606610 
.9607086 



41 

42 
4i 
443. 
45 



,9607561 
,9608036 
.9608512| 
,9608987 
,9609462; 

96099371 
9610412 
96108871 
,96113621 
9611837] 



No 



J16V3. 
9162*3. 
9163 3 
9164'S. 
9165 



9171 
9172 
9173 
9174 
9175 



9619429 
.9619903 
9620387 
9620851 
962132*19215! 



9021799 
,9622272, 
.9622746 
.9623220! 
,9623993 



3.9624167 
39624640 
9625114 
3.9625587 
39626061 



9176 3.9626534 
9177(3.9627007 
91783.9627481 
9179|3.9627954 
91 8039628427 



9181-3 9628900 
91823.9629373 
9183 ! 3.9629846. 
9184'3.9630319 
9185(39630792 



9186;3.9631264 
918713.963173; 
91883.9632210 
9189J3.9632683 
9190*3 9633155 



a 

3.96123*121 
; 9612887 
;.9613262 
.9613736' 
1.9614211 



9191*3,9633628 
919213 9634100 
9193 ! 3.9t>34573 
9194 : 39635045 
9195;3 9635517 



919613 9635990 
919713.9636462 
9198J3 9636934 
91993.9637406 
920013 9637878' 



9201 
9202 
9203 
9204 
9205 



9206 
9207 
9208 
9209 
9210 



Lo£. 



3-9638350 
3.9638822 
3.9639294 
3.9639766 
3.9640238 



y2jl ! u.yfcOiati7 
9252 ! 3.96623i6 
9253'3.9662826 
9254|3 9663295 
9255 i 3.9e63764 



39640710 
39541181 
3 9641653 
3 9642125 
3.9642596 



9256 3.9004233 
9257i3.966470; 

9258 3.9665172 

9259 3.V665641 

9260 3 9666110' 



9211'3 
9212 ! 3. 

9213 3. 

9214 3 
3. 



.9643U68 
.9643539 
.9644011 
96444S2 
9644953 



yiitol 3-^666579 

9262 3«956704£ 

9263 3.9667517 

9264 3.9667985 

9265 3.9668454 



9216 
9217 
9218 
9219 
9220 



9221 
9222 



39647780 
3.9648251 
92233 9648722 
3.964910S 
3 9649664 



9224 

9225 



9226 
9227 
9228 
9229 
9230 



9231 
9232 
9233 
9234 
9235 



9236 
9237 
9238 
9239 
6240 



9241 
9242 
9243 
9244 
924, 



9246 
9247 
924b' 
9241 
9250 



9045425 
.9645896 
.9646367 
9646838 
9647209 



9206,3-^006923 
9267 3.9669392 
9268'3.9669860 

9269 3.9670326 

9270 39670797 



9271,3^071266 
9272.3 9671734 
9273:3-9672203 
9274 3.9672671 



8.9660135 
3.9650605 
3 9651076 
3.965154! 
8.9652017 



3.9652488 
3.9652958 
3.9653428 
3.9653899 
3 9654369 



3.9654839 
3.9655309' 
3,9655780 

3.9656250 
3.9656720' 



8.9657190 
3 9657660 
3.9658130 
3.9658599. 
89659C69 1 



3 U059539 
39660009 
S.9660478 
3.9660948 
3 9661417 



No., Loir. 



9275 3.9673139 



9276|3 90/3607 

9277 3X674076 

9278 39674544 

9279 3.9675012 

9280 39675480 



9281 
9282 
9283 
9284 
9285 



3 9076948 
3.9676416 
3.9676884 
3.9677351 
3 9677819 



9286 
9287 
9288 
3289 
9290 



3.V078287 
39678754 
3.9679222 
3.9679699 
39680157 



9291 
9292 
9293 
9294 
9295 



j l 2b6 
)297 
329ft 
9299 
9300 



3 9080625 
3 9681082 
3968155V 
3.9682027 
3 9 682494 

3.908a9bl 
3968342* 
3.968589; 
3.9684365 
^68482* 



344 



taGARITHMICK 



No. | Lo£. I|V«>.| Lojr. , 
yaui 3 yo352yo |9iUi|3.y/ U858T 

5302 3.9685763 ! 9352 3 9709045 



9303 3 9686230 

9304 3.9686697! 



9305 3.9687164 t | 9355 l 3 97 10433 
93Uo 39687a30 '9356 3.9? 10 JO 2, 

9307 3.96880971 J9357.3 9711366 

9308 39638564 ,9358 3.971 1830! 

9309 3.9689030 '935^3.971 2294 ! 

9310 3.9689497 J9360 3.9712758! 



9311 3.9689903 9361 3 97132221 9411 

9312 3.9690430 |9362 3.9713686 9412 

9313 3.9690896 ,9363 3.9714150 9413 



9314 3.9691362 ;9364 3.9714614, 

9315 3 9691829 !9365a9715078 l 



9316 3.9692295, [9366 3.9715542 

9317 3.9692761; 9367,3.97160051 
£318 3.9693227l|9368 3.9716469 

9319 3.9693694 J9369 3.9716632 ( 

9320 3.9694159J 9370 3.97l?396 



9353;?.9709509 
93543.9709974' 



No. 



L<>Z 



9401 3.9731741 



9402 
9403 
9404 
9405 
L*4U0 
9407 
9408 
9409 
9410 



39732202 

3.9732664 

39733126 

.9733588 



9321 3.9694625 

9322 3 9695091 

9323 3.9695557 
93243.9696023 
9325J3 9696488 



'9371 3.9717859 
19372 3 9718323 
93733.9718786 
J93743.9719249 
937539719713 



9326139696954 
9327 3.9697420 
9328'3.9697885 
9329(3.9698351 
9330J3.9698816 

9331 



9376,3.9720176 
9377,3.9720639 
93783.9721102 
93793.9721565 
93303.9722028 



9332 
9333 
9334 
9335 



3.96U9282 
3.9699747 
3.9700213 
39700678 
3.9701143 



93813.9722491 



9382 
9383 
9384 
9385 



9336 3.9/01603 



9337 
9333 
9339 
9340 



3 9702074 
3 9702539 
3.9703004 
3 9703469 



9341 
9342 
9343 
9344 
9345 



9346 

934 

9348 

934? 

035C 



3.9703934 
3.9704399 
3.9704863, 
3.9705328: 
3.9705793; 



3.9706258 
3.9706722 
3.9707187 
39707652 
19708116 



3.9736358 
3.9736819 
3.9737281 

94143.9737742 

941513 97382031 

U416 

9417 

9418 

9419; 

9420 



J.9738664 
3.9739126 
3 973958; 
3.9740048 
3.9740509 



9421 
9422 
9423 
!9424 
9425 



3.9722954 
3.9723417 
3.9723880 

3 9724.343 1 
9386 3.9724805 
93873.9725268 
93883 9725731 

9389 3 9726193 

9390 39726656J 

9391 19727118 

9392 39727581 

9393 39728043 

9394 3.9728506 

9395 39728968 



9426 
9427 
9428 
9429 

9431 
9432 
9433 
9434 
9435 



9436 
^437 
9438 
9439 
9440 



No, 



9451 
945S 
9453 
9454 
9455 



3.9724050 
3 9734511 
3.9734973 
3.9735435 
£9735896 



9450 
9457 
9458 
9459 
9460 



Lc; 



3.9754778 
3.9755237 
3.9755697 
3.975615c 
5.975661 



3.9757075 
3.9757534 
3.9757993 
3.9758452 
3. 9758911 
946I13.9759370! 

9462 3.9759829! 

9463 3.9760288- 

9464 3.9760747! 
9465|3 9761206! 



'No. 



95ui; 

9502, 
9503 
9504 
9505' 



3 9777693 
3.9778150 
3.9778607 
3.977906* 
3.9779521 



9507! 
9508 
9509 
9510 



9511 
9512 
9513 
9514 
9515 



.9740970 
3.9741431 
3.9741892 
3.9742353 
3.9742814 



9406|3.9761665' 
9467 3.9762 124i 
9468J3.9762582 1 
9469 3.9763041; 
9470;3. 9763500 ! 

'9471J39763958! 
947^3.9764417j 
9473|&9764875i 
9474 ! 3.9765334 ! 
9475'3.9765792| 



3.9743274 
3.9743735 
1.9744166 
3.9744656 
3.9745117 



3.9745577- 
3.9746038 
3.9746498 
3.9746959: 
3. 9747419 J 
39747879' 
39748340* 
3 9748800! 
3.9749260; 
3.9749720' 



9476 3.9766251; 



9477 
9478 
9479 
9480 



3.9766709 
3.9767167 
3.9767625 
3.9768083 



9481 
9482 
9483 
9484 
94-85 

9486 
9487 
9488 
9489 
9490 



3 9768541' 
39769000 
3.9769458 
3.97699151 
3.9770377 



9441 
9442 
9443 
9444 



3.9750180. 
3.9750640! 
3.9751100. 
397515G0; 



93y6 3.9729439 
9397 3.9729892 
9393 3.9730354 

9399 3.9730816 

9400 3.9731279' 



94453.9752020: 



9446.3.9752479, 
9447 3.9752939: 
94483.9753399 
,9449 3 9753858; 
l9450 ! 3 97543 W 



3 9770831 
3.9771289 
3 9771747 
3.9772204 
3.9772662 



9491 
9492 
9493 
9494 
949, 



••;496 
•0497 

9498 



95C0fe 



3.9773120 
3.977S577 
39774035 
3.9774492 
3.9774950 



3.9775407 

3.977580* 

3.-9776322 

HQ? 3.9776779 

.9777236 



Log. 



3.9779978 
3.9780435 
3.9780892 
3.9781348 
3.9781805 



3.9782263 
3.9782718 
3.9783175 
3.9783631 
3.9784088 



9516 
9517 
9518 
9519 
9520 



9521 

9522 
9523 
9524 
9525 



9526 
9527 
9528 
9529 
9530 



9531; 

9532 



3.9784544 
3.9785001 
3.9785457 
3.9785913 
3.9786369 



3.9786826 
39787282 
3.9787738 
3.9788194 
3.9788650 



3.9789106 
3.9789562 
3.9790017 
3.979047: 
3. 979092 9 

3.9791585 

3.9791840 

9533, 3.9792296 

9534 3.9792751 

9535 3.9793207 



953613.9793662 
953*3.9794118 
9538 3.9794573 
9S39 , «3.97S50C8 
9540.J3.9795484 

y34li3.97959.W 
9542l|a9796S94 
9543 3.9796849 
)544nS 9797304 



9545 



9546 
9547 

19548 
!954<: 
'9550 



[3.9797759! 



5J.979WU 
'3.9798669 
^3.9699124 
3 9799579 
V V)800034 



ARITHMETICS. 



245 



9551 
9552 
955S 



3.9805033 
95623.9805487 



9561 

9562 

95633.9805942 

95643.9806396 

9565 S 9806850 



95663.98ur304 
95673.9807758 
.9808212 
.9808666 
3.9809119 



9568 , 3, 
95693. 
9570 



9571 
9572 
9573 



No. 



3.9800488 
3.9800945 
3.9801398 f 
9554 3.9801852 
95553.98023071 



95563.9802761J 
9557a9803216' 
95583.9803670 
95593.9804125! 
95603.9804579 



Log. 



9601 
9602 
9603 



3.9809573 
3.9810027 
3.9810481 
95743.98109341 
95753.9811388! 



9606 
9607 
9608 
9609 
9610 



9611 
9612 
9613 



3.9827686J 
96123.9828138I 

3.9828589 
96143.9829041 
9615 3.9829493 



9617 
9618 
9619 
9620 



9576'3.9811841 
95773.9812295 
95783.9812748 
9579J3 9813202 
9580)3.9813655 



9581|3 9814108 
95823.9814562* 
9583 3.9815015 
95843.9815468 
958513.9815921 



9586J3.9816374 
95873.9816827 
9588 ; 3.9817280 
958913.9817723 
9590(3.98181861 



9591 
9592 



3.9818639 
.9819092 



9593 £9819544 



9594 
9595 



S.9819997 
&98204J0 



9597 
9598 



9596 3.9820902 
3.9821355 
8.9821807 
9599 3.9822260 
96003.9822712 



No, 



3.9823165 
3.9823617! 
3.9824069 
96043.98245221 
9605 3.9824974, 



103. 



3.9825426: 

.9825878: 

3.98263801 

3.98267821 

98272341 



96163.9829945 
3 9830396 
3.9830848 
3.9831299 
3.9831751 



9621 
9622 3. 
9623. 



9626 
9627 
9628 
9629 



9631 
9632 
9633 



96i 
9637 



9639 
9640 



9643 
9644 3, 



Lo*. | 



965113.9845723 
9652 ! 3.9846173 
9653J3.9846623 
9654 ! 3:9847073 
9655J3.9847523 



9656,3 9847973 
96573.9848422 
965813.9848872 
9659,3.9849322 
9660i3 9849771 



9661.3.9850221 
96623.9850670 
9663|3.985U20 
9664'3^851569 
96653.9852019 



966639852468 
96673 .3852917? 
9668:3 9853366 
9669,3.9853816 
96703 9854265 



9832202! 

1.9832654, 

39833105 

9624 S.9833556 

9625 39834007- 



3.9834459, 
3.9834910: 
3.9835361' 
3.9835812 
96303.9836263 



3.9836714 
3.9837165 
3.9837616 
963439838066 
9635 39838517 



63.< 



.9838968 

3.9839419 

96383.9839669 

1.9440320 

3.9840770 



9641 39841221 
96423.9841671 
3.9842122 
1.9842572 
96453 9843022 



96463.9843473 
96473.9843923 
96483.9844373 
96493.9844823 
96503.9845273 

32 



frio.j L,0£. 



9701J3.9868165 
9702 39868613 
97033.9869060 
9704*3 9869508 
970513 9869955 



97113. 

9712|3 

971, 

97143. 

9715 



9671,3.9854714 
9672'39855163 
9673J3.98556I2 
9674|3.9856061 
9675J3.9856510 



V676 3.9856959, 
9677i3.9857407i 
9678[3. 9858560 
^679 3 9858351 
968039858753 



9681 
9682 
9683 
9684 
9685 



9686 
9687 



9691 
9692 
9693 



96?6 3. 
9697 



9699 
9700 



39859202= 
3.9859651; 
3.9860099; 
3.9860548 
.9860996: 



3 9861445! 

3.9861893J 
96883.9862341 

3.98627901 
96903.9863238 



3.9863686' 
96923.9864134 s 

39864582J 
9694 3.98653301 
96953.9865478; 



.98659261 

9866374! 

9698 3.9866822; 

1.9867270, 

3.98677171 



No*| Log. 



97063 
97073 
9708|3 
970913 
971013 



9870403 
.9870850 
.9871298 
.9871745 
9872192 



97563.9892718 
975739893163 
9758'3.9893608 
'9759.3.9894053 
!9760|3.9894498 



97163.9874875 
97173.9875322 
.9875769 
.9876216 
9876663 



9718 
9719 
9720 



9721 
9722 
9723 



39877109 
3.9^77556 
3.9878003 
97243.9878450 
9725 3.9878896! 



9728 
9729 



9733 
9734 
97353 



9737 
9733. 
9739 
9740 



9872640 
9873087 
.9873534 
9873981 
9874428 



197013.9894943 
!9762j3.9895388 
9763 3.9895833 
,976413.9896278 
i9765'3.9896722 



97263.9879^43 
97273.9$79789 
3.9880236 
3.9880682 
97303.9881128 



97313.9881575 
97323.9882021 
3.9882467 
.9882913 
9883360 



97363.9883460 

3.9884252 

3.9384698 

3.9865144 

9885590 



9741 3.9886035 
9742.3.9886481 
97433.9886927 
9744I3.9887373 
9745 3.9887818 



9746 398S8264 
97473.9888710 
974*3.9889155 

9749 3.9889601 

9750 3.9890046) 980O 1 



No. | Log. 



97513.9890492 
9752.'3.9890937 
97533.9891382 
9754 r 3. 9891828 
9755(3.9892273 



97663 9897167 
976739897612 
97683.9898057 
3.9898501 
9898946 



9f69 
9770 



,9771 
9772 
977^ 
19774 
'9775 



9776 
,9777 
9778 
9779 
9780 



|9781 



1.9899390 
3.9899835 
3.9900279 
3.9900723 
39901168 



3.9901612 
39902056 
.9902500 
3.9902944 
3.9903389 



J.9903833 



97823.9904277 
9783 3.9904721 
97843.9905164 
9785 3.9905608 



97863.9906052 
978759906496 
9788 3.9906940 
97893.9907383 
9790 3 9907827 



979139908271 
97923.9908714 
9793'3.9909158 
97943.9909601 
979539910044 



9796 3.991 L>48* 
979713.9914931 
9798.3.9911374 
9799 3.991181K. 
te ,$912ag 



846 



logarithmiCk 



\»* 



9601 i.99127u4 9851 3.9934803; 

9802 3.<,>913147j!98523.9935244 

9803 3 9913590 9853 3.9935685 
98043.9914033 98543.9936125 
9805 3.9914476 .9855J39936566 



9806 
9807 
9808 
9809 



3.9914919 

,9915362 

9915805 

3.9916247 

98103 9916690 



9811 
9812 
9813 
9314 



Loq\ !,No. Log. 



99013.9956791 
9902 3.9957229' 
990339957668 
99043.9958106 
9905 3.9958545 



J98563 9937007J 
!9857!39937447J 
9858:39937888! 
!9859'3.9938329 
!9860 J 39938769 



99063.9958983; 
99073.9959422 
99083.9959860 
9909 3.9960298 
99103 9960736 



3.9917133 •986i;3.9939209j 
3 9917575 '9862,3 9939650. 
39918018-9863 3.9940090 



3 991846119864 
981 5|3 9918903*9865 



981613 9919345; 
9817J3.9919788, 
98183.9920280: 
9819139920673 
9820'3 9921115 



9866 
986/ 



9821-3.9921557 

9822 3 9921999 

9823 3.9922441 
982413.9922884 
982539923326 



39940531 
39940971 



3.9941411 
i.9941851 
9868J3 9942291 
9869 3.9942731 
9870S994S171 



9871 
9872 
J987: 
|9874 
f9875 



3.9943611 
3.9944051 
2.9944491 
3.9944931,|9924 
9945371 



9826 3.9923768 
98273 9924210 
9828,3.9924651 
9829:3.9925093 
98303.9925535 



i987#|3.9945»ll , 
3 9946250 
3.9946690 
39947130 

3.9947569 



9877 
9878 
19879 
J9880 



9831 3 9925S77< 

9832 3 9926419 

9833 3.9926860 
9834*3.9927302 

9835:3.9927744 



,9881 
9882 
19883 
19884 
19885 



98363.9928185 
9337 i 3.9928627 
9838 3 9929068 
9839,39929510. 
98403 9929951 



9889 
9890 



9b41!3.9930o92 
9842]3 99303*4 
9843 3.99^1275 
98443.9931716 
9S45i 10932157! 



9K4o 3 9932 J9& 
984*3^53039: 
9848;.3 9?334fi0 : 
9849'fV^33921'i 
9850 i 3>J343«2l 



9891 
19892 

19893 
9894 

J9895 



98*6 
9897 

98'*' 
93'"/<- 
990Ul r 



9911 3*961175 
99123.9961613 
9913 3.9962052 
991439962489 
9915 3.9962927 



39948009 
3.9948448 
3.9948888 
3 9949327 
3.9949767, 



5 9886 3.9950206 

9887 3.995*0645 

9888 39951-85; 
3.9951524» 
3.9951963! 



No.; JLo^. 



9916 3.9909365 
9917*3.9963803 
99183.9964241 
99193.9964679 
99203.9965117 



99213.9965554 
9022;3.9965992: 
9923 3.9966430: 
3.9966867i 
9925 3.9967305; 



V9263.f967743 
9927-3.9968180 
9928;3 9968618! 
9929 3.9969055! 
99303.9969492 



No. I Lotr. 



9951 3.9978667 1 

9952 3.9979104 

9953 3.9979540 
99543.9979*976 
99543 9980413 



995639980849 
9957;3.9981285 
995839981721 
99593.9982157 
6960;3|>982593 



9961.39983029 
9962S3.9983465 
9963J3.9983901 
9964i3.9984337 
9965 3 9984773 



9966 t o.9985209 
9967J3.9985645 
9968'3.9986680 
9969*3.9986516 
99703 9986951 



39987387 
99723.9987823 

3.9988258 
9974*3.9988694 

39989129 



9971 
9972 
6973 
9974 
9975 



9976 
9977 



3.9989564 
3.9990000 

99783.9990435 
3.9990870 

998039991305 



99313.9969930:! 998139991740 
99323.9970367) 99823.9992176 
99333.9970804!; 9983 3.9992611 
993439971241! 9984| 3.9993046 
99S53.9971679J| 



9936i3.9972ll6i 
9937iS-997255a 
99383.9972990 
9939 3 93734271 
9940 ! 3.9973864i'. 



^9863.9993916 
99873.9994350 
9988 ! 39994785 
998913.9995220 
99903.9995655 



9952402 
9952841 
P953280 
9953799 
9954158 



.9954597 
.9955036 
.9955474 
9955913 
.9956352 



99413 9974301,! 99913 
99423.9974737| 9992*3 
99433 9975174' 9993 3 
9944-3.^75619 9994J3 
9945 3 9976048 9°9r m 



5J3 



9996089 
9996524 
'9996959 
9997393 

9997828 



9946,3 9976484 : 999613 -9998662 
9947 3.9976921 j« 9997 3.9998697' 
994a3 9977358ij 9998 3-9999131 
9949}3.9977794t| 9999 3-9999566 
9950 , 3.9978231 ''lOOOO 4.0000000 



APPENDIX. 

THE hyfierboliek curve was found also, by Lord Napier, the 
celebrated inventor of L.Q$arithma y to afford another source for 
developing and illustrating the properties and construction of 
logarithms. ' For the hyperbfliick areas lying between the curve* 
and one asymtote, when they are bounded by ordinates parallel 
t6 the other asymtote, are analagous to the logarithms of their 
abscissas or parts of the asymtote. And although Napier's 
logarithms are commonly said to be the same as hyperbolick. 
logarithms, it is not to be understood that hyperbolas exhibit 
Napier's logarithms only, but indeed all other possible systems 
of logarithms whatever. For it has been found that the loga- 
rithms of the same number in two^ different systems are to one 
another as the reciprocals of the logarithms of the radical num- 
bers of those systems, these latter logarithms, beings taken ac- 
cording to. any system whatever t 
Thus* 

Brigg*s logarithm of any number ; 

is to JS/afiier-n logarithm of the same number: r 

as JBrigg's logarithm of 1Q.\ 

is to Na/iier's. logarithm of 10 : 
But Brigg's logarithm, or the common logarithm of 10 is 1 , 
and Napier's logarithm of 10 is 2^3Q2585693; whence if Brigg's> 
or the coinmoa logarithm of any number, be denoted by c. l, 

• a 

" Note. This curve greatly facilitates the conception of loga* 
rithms to the imagination, arid affords almost an intuitive proof, 
of the very important property of -their Auctions, or very small 
increments, viz. that the Auction of the number is to t&e. Auction 
of the logarithm, as the number is to the subtangent... And it is 
evident, that in the beginning of the generation of th«bc areas. 
from the vertex of the hyperbola, the rascent increment of the 
ahscisse drawn into the altitude I, is to the increment oi" the- 
area,. as radius is to the angle of the ordinate and abscisse, or of. 
the asymtotcs : and at the beginning of the logarithms, the ra- 
ftcent increment of the natural numbers is to the increment of 
tne logarithms as 1 is to the modulus of the system. 
. Hence, we easily discover, that the angle formed by the asym- 
totes of the hyperbola, exhibiting Brigg's System of Loga- 
rithms, will be 25° 44* 25" ; this bting the angle whose bine h 
Q.43429443I9, &c. the modulus of the common system. 






Hi LOGAMTHMICK 

and Napier's or the hyperbolick logarithm of the same nufnber> 
by b. l. we shall have 2.302585095 : 1 ::*.£.: c. t. ; or 

1 

X. L. X X.3\Mi>B*uy5 ~ h. l. X .4342944819 = c. x. as was re* 
Quired. 

In comparing the different systems of logarithms, Lord Na- 
pier's is evidently the mo*t simple in respect of facility of con- 
ttruction, because its modulus is unity.* Thus, suppose it 
were required to find the Napierean logarithm of the number 2+ 

Here employing the formula. 

log. 2 = 2 51 -+ J-. + JL + J- + J_-f &c. I 

= A-i |Bf }C f-;Df{E &c, where A is put for 

f B for| A, C foi i B, D for £ C, &c. 
The calculation wiil be as follows: — 

A — y = .666666666 A = .6G6666666 
B = $ A =* .074074074* B = .02469 13S8 



C = JB' = 


.008230453 fC c= 


.001646091 


D = *C = 


.000914495 4 D = 


.000130642 


E=s= *D = 


.000101611 £E = 


.000011290 


F=>£E = 


.000011290 /- f F = 


.000001026 


G = | F = 


.000001 254 tV G3BS 


.000000096 


H = £G = 


.000000139 «VH=« 


.000000009 


I = fcH== 


.600000015 jV 1 — 


.000000001 



Nap. log. 2 = .693144179 
Or, retaining only eight figures* Nap. log. 2 = .69314718 
Having obtained the log. of 2, we can easily find the- loga- 
rithm of 4, 8, and in general of any power of 2. 

Ex. 2. Required the Napierean logarithm of 5. • * 

By employing the same formula as before, and proceeding; 

exactly as in the last example, by taking the sum of a sufficient 

* Note. Mr. Baron Maseres gives the following definition of 
tfce Modulus) namely, " that it is the limit of the magnitude of a 
fourth proportional to these three quantities viz. the difference 
of any two natural numbers that are very nearly equal to each 
other, either of the said numbers and the logarithm or measure 
of the ratio they have to each othet." Or we may define the 
modulus to be the natural number at that part of the system of 
logarithms, where the Auction of the number is equal tothc 
fluciion of the logarithm, or where the numbers and logarithms 
are have equal differences. And hence it follows, that the log- 
arithms of equal numbers, or of equal ratios, in different sys- 
tems, arc to one another* as the moduli of those systems. 



ARITHMETIC*: $4* 

number of the terms of the series, we shall find the Napiereafc 
log. of 5 =J= 1.609437912. 

The Napierean logarithms of 2 and 5 being found* the Nap. 

log. of 10 = 2X5 becomes known. 

Thus, to log. 2 «' . 693J47179 

Add log. 5= 1.609437912 

T|tf5 sum is log. 10 = 2.302585091 

Or, retaining eight figures, log, 10 == 2.30256509. 
Whence also the modulus of the common system of loga- 
rithms is known, for it is the reciprocal of the Napierean loga- 
rithm of 10, or — ~- a .434294482 

' 230258509 

We can now easily find the common logarithms of the num- 
bers 2 and 5 ; for yre have only to multiply the Napierean log. 
already found by the modulus .434294482, or divide them by 
its reciprocal 2.30258509, and the products, or quotients, arc 
the logarithms sought. 

Thus retaining only seven decimal places of the products, we 
have, 

Com. Log. 2 = .69314718 X .4342944=** .3010300 
Com. Log. 5 = 1.60943791 X ,4842944 s .6989700 
Com. Log. 10 =ss 2.30258509 X .4342944 = 1.0000000 
Or, the Common Logarithm may be found by putting M for 
.4342944, as in the following expression. 

C5 3^5* 5.5* 7.6* > 

And the calculation will stand th«s j 
Log. 2 = 30103000 

~= .17371779 
5 

2M 
3.5* 

2M ■ A 

-j= .0000055$ 



.0023 1684 



5.5* 

•2M 

^-j5= .00000159 

2M 

gJS** .00000005 



Com.Log. 3= .47712126 
And .47712126 *t- .4S429448 m Nap. log. 5=» 1.096SC9&. 



2S0 LOGARITHMIC* 

f 

Another Method, to find the Logarithm of any of the natur- 
al number*, 1, 2, 3, 4, Sec. 

* RULE.* 

1. Take the geometrical series, 1, 10, 100, 1000, 10000, 8cc. 
mad apply to it the corresponding arithmetical series 1,2, 3,4, 
&c. as logarithms. - . ' 

2. Find a geometrick mean between 1 and 10, 10 and 100> 
or any other two adjaceaf terms of the series betwixt which the 
proposed number lies. 

S. Between the mean, thus found, and the nearest extreme, 
find another geometrical mean, in the same manner ; and so on, 
till you are arrived within the proposed limit of the number 
whose logarithm is sought. 

4. Find as many arithmetical means, in the same order as you 
found the geometrical ones, and the last of these will be the 
logarithm answering to the number requered. 
fxamples. 

Let'it be required to find the logarithm of the number 9. 

Here the numbers between which 9 lies are 1 and 10. 

Firit, then, the logarithm of 10 is 1, and the log. of 1 is ; 

therefore _i_ =* ,5 is the arithmetical mean, and ^(1X10) 

3 
«^/io« 3. 1622777= geometrick mean 1 whence the loga- 
rithm of 3.1622777 is .5 

Secondly, the log. of 10 is 1, and the log. of 3.1622777 is 5 ; 

therefore "*"- = .75 = arithmetical mean, and */ (1.0 x 

3.1622777)=* 5.6234132 = geometrick mean ; whence the log* 
of 5.6234132 is .75. * 

Thirdly, the log. of 10 is 1, and the log. of 5.62341,32 is .75 * 

1 mlm 7 1 

therefore " r * — =.875 = arithmetical mean ; and */ (10 K 

z 
5.6234132) = 7.4989421 = geometrick mean : whence the log. 
of. 7. 4989421 is .875. 

Fourthly, the log of 10 is 1, and the log. of 7.4989421 is .875 ; 

1 I &7< 

therefore -. T '. . = .9375 = arithmetical mean, and v'OO x 

2 

7.4989421)= 8.6596431 = geometrick mean : whence the log* 
of 8.6596431 is .9375. 



* Note. The reader who wishes to inform himself more par- 
ticularly concerning {he history, nature, and construction of 
logarithms, may consult Hutton's Mathematical Tables, pub- 
lished a few years sinc<?> where he will find his curiosity am* 
ply gratified. 



ARITHMETICK 2*1 

fifthly, the log.of lOis 1, and the log. of 8.6596431 is .9375; 

therefore 1+ * 9375 -= 96875 = arithmetical mean, and V^IO* 

2 
8.6596431) «= 9.3057204 = geometrick mean : whence the log. 
of 9.3057204 is .96875. 

Sixthly, the log. of 8.6596431 is .9375, and the logarithm of 

9.3057204 is .96875 ; therefore ^ilL±-^£L 5 =.953125 «= 

2 

arithmetical mean, and ^(8.6596431 X 9.3057204) = 8.976*713 

*■ geometrick mean : whence the log. of 8 976B713 is .953125 

And, proceeding in this manner, after 25 extractions the log- 
arithm of 8.9999998 will be found to be .9542425 ; which may 
be taken for the logarithm of 9, hecause it differs from it onlv 
by io66066 > and is therefore sufficiently exact for all practical 
purposes. 

And in the same manner were the logarithms of almost all 
the prime numbers found by Lord Napier ; a work so incredibly 
laborious, that the unremitted industry of several years 
scarcely sufficient for its achievement. 



j 



The reader in requeued to correct the following which arc 
the fifinciftal 

ERRATA. 

JPage 15, 26tb line from the top, for 126 gals, read 128. 
16, 9 for £136,58 read £163,07 

26, 27 for g 428,50 read 8 429.50 

27, 28 for 182 yds. read 132 yds, 
32,26 for 12114 % %\ read 1 270{4$ 
33, 8 for £481 read £431 

40, 33 for 15* read 3* 

41, 3 for 12 Ik 2 m. read 1 1 b. 58m. 

ibid, t for 5 o'clock 19m. 37sec. read 6 o'clock 

40m. 23, sec. 
ibid: 14 for g 1940 read 8 1946 
ibid. 34 for 8 50 read g 5 1 . 
67, 26 for 8 975,99cta. read g 1042,86 ct&. 
7*3, 26 for 73% gal. read 7£ gal. 
ibid. 32 for 22 lb. read 2 lb. 
74, 26 erase -2d. 

ibid. 42 for ad infinetum read ad infinitum. 
77,15 for Duble read Doublb. 
79, 24 for fully read briefly. 
84, 11 for 3, the root read 3758, the root. 
87, 18 for square read cube. 
95, 7 for *From the square of the height, subtract 
the height, read * From the square of the distance divided by 
the height, subtract the height. 

110, 19 for £ 1 read£7 * 

134, 9 for 64 read 40 
143—147/*. for minuets read minutes. 

168 11 for South-Descending read 6 Signs South- 

(Descending. 
191, 26 for North Ascending read North Descend- 

201, 16 for right read left. \ (ing. 

202, 13 erase totally. 



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